No. He gets two beams.
There is actually no such thing as 'caster level' in 5E (in contrast to previous editions).
In 5E, there is just the spell slot you use to cast the spell.
The single exception is casting a cantrip. For cantrips, you use your total character level, as opposed to 'class level in the class that gave you the cantrip' (because you can get cantrips without ever having a level in a spellcasting class) or 'caster level' (which doesn't exist in 5E).
Your pal 3/war 2 is a 5th level character, and so gets two beams.