Sword of Spirit
Legend
Traditionally, the bladesinger wields a longsword, wears chainmail (best represented as a chain shirt for 5e), and aspires to wear the coveted elven chain, which is designed with them in mind more than any other archetype.
5e doesn't really let you do this. I'm going to present the premises for discussion, explain the problem, and then discuss the various house rule options I've already thought of and why they won't really fix the problem in a satisfying way.
Premises
Premise 1. The game should not punish you mechanically for playing to lore. Sure, occasionally the designers might find themselves in a difficult place with rules interaction, so maybe you can make a minor sacrifice (a point of AC or damage, etc) to represent the traditional lore, but it shouldn't be a significant sacrifice, because that actively punishes/discourages you from playing to the lore.
Premise 2. Medium armor is only a good option for a Strength-based warrior that lacks heavy armor proficiency. A Strength character with heavy armor proficiency would lose AC and gain nothing (they'd lose even more AC if they wanted to avoid a Stealth penalty) by wearing medium armor. A Dex based character would lose good Stealth, and the only they would gain would be a bit of AC at low level only by wearing medium armor. Otherwise, wearing medium armor is a penalty for them. (Sub-premise: An armor category that is only useful at a minority of the level range is a problem.) Even a Strength character has to invest a 14 in Dex as a tertiary stat in order to get the maximum AC from medium armor (which is worse than the AC from heavy armor with no investment). Therefore only a Strength-based warrior that has proficiency in medium armor, but lacks proficiency in heavy armor (generally a Strength based barbarian or ranger), gets any lasting benefit from medium armor, and even that is kind of expensive.
Premise 3. There is no assumption that a character can plan to ever get a specific magic item. Treating such a specific item as an essential part of your character doesn't work. So you cannot design a character whose realization requires the acquisition of a particular magic item. You can design one for whom such an item would be an awesome and desirable supplement, but not a requirement.
Considerations
There are a number of rules points that I'm just going to list so everyone can have them accessible.
A) Longswords are not finesse weapons, therefore they require a Strength-based wielder.
B) You cannot cast spells in armor you are not proficient with.
C) Bladesingers gain proficiency with light armor, but not with medium.
D) Bladesingers' defining feature that enables their melee presence, Bladesong, cannot be used in armor heavier than light, proficient or not
E) Elven chain is a medium armor
F) All characters are treated as if they were proficient in elven chain
The Problem
Because of the lore and the premises, a bladesinger needs to be a Strength-based medium armor wearer; but they are not proficient in medium armor, nor could they use Bladesong in it if they were. Their mechanical design therefore supports a Dex-based light armor wearer instead of the traditional longsword and mail (chain shirt) or elven chain character.
Possbilities for House Rules
This, is not easy. I should point out that I like simple and elegant house rules. The kind you can write in a single non-convoluted sentence. Here were the ones I considered, and why they failed.
#1) Bladesingers can use Bladesong and cast spells in elven chain. The main problem with this one is that it violates Premises 2 and 3. They can't wear regular chain shirts, so they are hanging everything on the possibility of getting that elven chain, while suffering mechanically as a Strength-based light armor wearer in the meantime.
#2) Elven chain becomes light armor. This one hurts more than it helps. It lets Bladesingers wear elven chain, but either they are a Dex-based character contrary to lore, or they are a Strength-based character in anticipation of getting elven chain, in violation of Premise 3. It also makes elven chain a good buff for other Dex-based characters, which isn't really a desirable outcome, and fails to give Bladesingers any particular benefit from it. Worse than option #1.
#3) Bladesingers can use Dexterity with longswords. I'm just bringing this one up since I'm sure someone will think of it. By itself it does nothing for the armor, but even if you combine this with #2 (starting to get into convoluted territory), it would still keep them away from chain until and unless they found elven chain.
#4) Bladesingers gain medium armor proficiency and can use Bladesong with it. On the surface this seems to fix the problem. They can wear medium armor from the start, so they stay lore friendly, and once they find elven chain they are fully embodying their archetype. The problem with this is that elven chain grants them no particular benefit that would make it a special coveted armor for them. They're better off with a +1 breastplate. In contrast to the lore that makes elven chain the jelly to Bladesinger's peanut butter, it is actually better for a non-Bladesinger wizard, sorcerer, lore bard, or Blade Pact warlock. This is the best house rule option, but it is still unsatisfying.
Who has a better house rule option for me?
5e doesn't really let you do this. I'm going to present the premises for discussion, explain the problem, and then discuss the various house rule options I've already thought of and why they won't really fix the problem in a satisfying way.
Premises
Premise 1. The game should not punish you mechanically for playing to lore. Sure, occasionally the designers might find themselves in a difficult place with rules interaction, so maybe you can make a minor sacrifice (a point of AC or damage, etc) to represent the traditional lore, but it shouldn't be a significant sacrifice, because that actively punishes/discourages you from playing to the lore.
Premise 2. Medium armor is only a good option for a Strength-based warrior that lacks heavy armor proficiency. A Strength character with heavy armor proficiency would lose AC and gain nothing (they'd lose even more AC if they wanted to avoid a Stealth penalty) by wearing medium armor. A Dex based character would lose good Stealth, and the only they would gain would be a bit of AC at low level only by wearing medium armor. Otherwise, wearing medium armor is a penalty for them. (Sub-premise: An armor category that is only useful at a minority of the level range is a problem.) Even a Strength character has to invest a 14 in Dex as a tertiary stat in order to get the maximum AC from medium armor (which is worse than the AC from heavy armor with no investment). Therefore only a Strength-based warrior that has proficiency in medium armor, but lacks proficiency in heavy armor (generally a Strength based barbarian or ranger), gets any lasting benefit from medium armor, and even that is kind of expensive.
Premise 3. There is no assumption that a character can plan to ever get a specific magic item. Treating such a specific item as an essential part of your character doesn't work. So you cannot design a character whose realization requires the acquisition of a particular magic item. You can design one for whom such an item would be an awesome and desirable supplement, but not a requirement.
Considerations
There are a number of rules points that I'm just going to list so everyone can have them accessible.
A) Longswords are not finesse weapons, therefore they require a Strength-based wielder.
B) You cannot cast spells in armor you are not proficient with.
C) Bladesingers gain proficiency with light armor, but not with medium.
D) Bladesingers' defining feature that enables their melee presence, Bladesong, cannot be used in armor heavier than light, proficient or not
E) Elven chain is a medium armor
F) All characters are treated as if they were proficient in elven chain
The Problem
Because of the lore and the premises, a bladesinger needs to be a Strength-based medium armor wearer; but they are not proficient in medium armor, nor could they use Bladesong in it if they were. Their mechanical design therefore supports a Dex-based light armor wearer instead of the traditional longsword and mail (chain shirt) or elven chain character.
Possbilities for House Rules
This, is not easy. I should point out that I like simple and elegant house rules. The kind you can write in a single non-convoluted sentence. Here were the ones I considered, and why they failed.
#1) Bladesingers can use Bladesong and cast spells in elven chain. The main problem with this one is that it violates Premises 2 and 3. They can't wear regular chain shirts, so they are hanging everything on the possibility of getting that elven chain, while suffering mechanically as a Strength-based light armor wearer in the meantime.
#2) Elven chain becomes light armor. This one hurts more than it helps. It lets Bladesingers wear elven chain, but either they are a Dex-based character contrary to lore, or they are a Strength-based character in anticipation of getting elven chain, in violation of Premise 3. It also makes elven chain a good buff for other Dex-based characters, which isn't really a desirable outcome, and fails to give Bladesingers any particular benefit from it. Worse than option #1.
#3) Bladesingers can use Dexterity with longswords. I'm just bringing this one up since I'm sure someone will think of it. By itself it does nothing for the armor, but even if you combine this with #2 (starting to get into convoluted territory), it would still keep them away from chain until and unless they found elven chain.
#4) Bladesingers gain medium armor proficiency and can use Bladesong with it. On the surface this seems to fix the problem. They can wear medium armor from the start, so they stay lore friendly, and once they find elven chain they are fully embodying their archetype. The problem with this is that elven chain grants them no particular benefit that would make it a special coveted armor for them. They're better off with a +1 breastplate. In contrast to the lore that makes elven chain the jelly to Bladesinger's peanut butter, it is actually better for a non-Bladesinger wizard, sorcerer, lore bard, or Blade Pact warlock. This is the best house rule option, but it is still unsatisfying.
Who has a better house rule option for me?