This should work, yes.
Balance-wise, it seems good but not broken to me. In order to use this trick, you have to cast one of the [xxx] smite spells, which have substantial drawbacks compared to using the same spell slot for regular paladin smite: You have to cast the spell in advance rather than when you hit, it costs a bonus action and concentration, and you can't fish for a crit and then dump your highest-level slot into it. And, of course, you are using your background feat.
In exchange for these limitations, you get a respectable amount of bonus damage, at the cost of a self-heal resource that you, a melee warrior, really need.