Take the above comment with a grain of salt, as I've yet to see anyone in a bathing suit at a signature, or even the nicer 1TS, except for one occasion at Yachtsman when the kids clearly must have put up a fight about changing as the parents were dressed very nicely.
I've never seen anyone in bathing suits at signatures on any of our trips and the only place I've seen tank tops/ball caps was Le Cellier. We've done 29 signature restaurant visits since 2010 and about 25 1TS lunch time meals.
Someone said plenty of others will be misbehaving, so I guess we've just been lucky on our nearly 30 signature experiences in the past couple years that we've encountered ONE child acting out and that was in Flying Fish
As long as people are presentable and well-mannered, I could care less if they have cargo shorts and a ball cap. I don't tend to stare at other diners at dinner for it to 'bother' me or hinder my experience. I'd rather a quiet, polite individual in a backwards ball cap sit next to me than a portly, loud snob in a suit with his wife pouring over the sides of her seats with whiskers in her chin, like the experience we had at Flying Fish next to a couple who was wealthy, dressed well, but completely ignorant and unsightly.
Anyway, to the OP. Children in park clothes is fine and common place. I've seen plenty of families dressed well and some in suits who've still allowed their children to 'relax' and 'be a kid' by wearing their mickey mouse shirt and light up sneakers. In Jiko, Artist Point, Yachtsman and Narcoossee's park clothing is the vast minority, but you're not going to feel 'out of place' unless you're really bumming it, like hill billy jean shorts and crab trap sandals and a park-sweat-stretched t-shirt.
With children, ultimately as long as they behave, they can wear anything without people paying any mind. I'd rather your whole family be in tuxedo t-shirts and swimtrunks with quiet children than a family in full-formal attire with children acting out, whining and pounding silverware.
Le Cellier and Hollywood Brown Derby are the most relaxed, with the most amount of patrons in park attire as they're located in parks.
Flying Fish would be the next most relaxed, as it's a more lively, bright and 'fun' atmosphere than it is romantic/fine dining, but the food and service is excellent. A large mix of park/business casual/formal dress here.
Jiko has mostly business casual with SOME park attire mixed in
Narcoossee's, Artist Point, Yachtsman tend to have very few guests in park attire, with Narcoossee's having the more suits in the restaurant than the other two that we've noticed.
Shula's is the most formal, second to Victoria and Alberts (heavily enforced dress code).