Here's my two cents- you really seem to be making a big deal about how your daughter loves her middle name. Most 4 year olds don't even know their middle name let alone was that attached to it. It seems like maybe you are projecting a bit. I agree with your wife that two middle names are cumbersome. Your daughter is still so young, she won't remember the name change. If she was 12 or something, that would be different.

It boils down to your wife grew these children inside of her, she had to deal with all the changes to her body, make sacrifices, birthed them, and they have your surname. Be understanding and let her have the middle name. Honestly, I think she will resent you forever if you don't.