I absolutely adore the Nimzo-Indian against 1.d4, but I simply cannot stand it when White plays 4.Bd2. What are some good ways to get an edge here? Should I play differently to what I would've if White had played e3 or Nf3?

4 Bd2 isn't a bad move, but it isn't a serious attempt for advantage, either.  Black is okay already, and how he proceeds depends upon personal taste.

Viable 4th move alternatives for Black include ...c5, ...d5, ...b6, and ...0-0.  Any of these are fine for Black.

Estragon is right; 4.Bd2 is a developing move, so how can it be bad? When you're playing Black, it's tempting feel that every time White deviates from theory we should be able to "punish" him. But chess doesn't work that way. Often these "inferior" moves are quite playable, and the best we can hope for is to reach equality more easily than in the main lines.

Having said that, I want to add that when your opponent plays an inferior move in the opening, you should go in high alert: this may be the first of a series of inferior moves, so you should be looking for opportunities yet to come.

You could compare it to playing like 1. e4 e5 2. Be2. Of course the move Be2 isn't directly bad but it's not very challenging either and black can finish hes development without any larger worries and can enjoy and equal game.