How to change entity texture within game?

In a way, yes. However, not a very practical way. You would need to create a mob for each way of dying it. Then make a procedure that if the mob is right clicked on, it would check if the player has dye in their hand. If the player does, it would kill the right-clicked mob. Then, it would spawn the according dye.

However, it just makes more sense that Pylo should just add some kind of dying option.