Just wondering why you require her to wear socks? I hate socks and never wear them inside. Most of my dck take their socks off pretty soon after arrival.
You mentioned picking battles but wearing socks doesn't seem worthy of a battle IMO. I have no problem letting a 4 year old choose what they want to wear. A 4 year old wailing and needing a blankie would concern me more than lack of socks.
Any reason the socks can't stay with her shoes to be worn for outside time? Maybe slippers would work.
If your issue is lack or support from the family, that's an entirely different story.