I've had an extremely strong willed 3 year old girl who wore skimpy princess dresses to daycare for an entire year including through a Melbourne winter. And we walked to and from daycare. I fought and I battled and I insisted and I didn't negotiate. And after being late to work for 2 weeks while this went on I gave up and gave in.
Eventually when it was 2 degrees she'd wear a light cardigan over her shoulders and if I was very lucky, leggings.
I endured every comment about how she'd get sick and freeze to death, and people would tutt tutt as I walked along in my overcoat with her in her princess frock.
In the end she finally started to wear normal clothes of her own accord. She never got sick, and to this day just doesn't really feel the cold (she's 8.5 now).
Being strong and having battles night after night wears you down and gets depressing. The experts say it takes a week or whatever but with my dd she never gave in. Not on this. For whatever reason in her tiny 3 year old brain it was really important to her so I let her win that one.
And now she's an incredibly rational kind and non argumentative child (most of the time). Winning this battle didn't turn her into a monster.
So I guess what I'm saying is assess how important is it to you to win this one and take it from there. I've not had this issue with my other kids only DD1. I share your pain. It's tough.