My DS' school has own-trousers-day once a week every week, own clothes sometimes for a full week most terms, and quite a lot of own-clothes-or-pyjamas fun days as a class reward. Last week we had a week of sports and sports wear (mainly just PE kit to be honest with a sports t-shirt and some shorts the other days).
I gather there's the odd silliness about sensible clothes and especially shoes among older children, but I have never heard of bullying. And all the parents can tell which DC in my DS' class has 6 older siblings and is wearing the uniform they all wore. Not such a great leveller.
Normalise non-uniform and everyone will get bored of it and everything will be fine.
I find it impossible to get second hand uniform (DS is uncomfortable in non-cotton clothes and the trousers and jumper are both slightly unusual colours - basically not black and navy it seems is unusual in our area!) so I'd love to be able to get a load of second hand non-school clothes and send him in jeans or joggers and NOT WHITE TOPS every day. He could wear his funky t-shirts I've made (that are NOT WHITE) every so often and he'd be happy too.
And we wouldn't have the battles over socks (I can't find any school socks, I'm going to wear these. No you aren't, have you looked? Yes I have. Well obviously not with your eyes).