And the cynic in me is wondering why he wanted to discourage you from going…because it kinda (very much) sounds like this is what he did
Absolutely.
It was "you're not dressed up enough" - to go to a fucking gym/pool 🙄
Then - when op pointed out there was nothing bad about her clothes, they were presentable - it was 'your hair's not done".
Again , whose hair needs to be "done" to go & watch a kid swimming in the spectator section of a gym pool?!
It was excuses, excuses, excuses.
Same as the BBQ - again they are pretty casual events; who would not take their gf to a BBQ because she didn't have a fancy dress etc to wear? Absolute bollocks.
He didn't want you at that BBQ for some reason and he didn't want you at that gym today for some reason.
Personally I'd guess that reason is related to the fact that he's a cheater.
(A cheater who gas lit you that you weren't official so it wasn't really cheating - that's why he told you clearly you two weren't official or exclusive & you were free to go out with & shag other guys too, right?! Oh he didn't? What a surprise).
He focuses on and criticises your looks/presentation because he knows that works. He knows he can make you feel inadequate and inferior.
He knows you'll bow down to that, instead of telling him to fuck off.
He knows there are no bad consequences for him from him doing that to you; you kept seeing him and shagging him after the BBQ thing, you'll be there when he comes back after this pool thing today, maybe a little but pissed off but nonetheless, he'll still get his dinner, get sex when he wants it, have a faithful partner at home etc. etc
It's a tactic that works for him to keep you down and to get rid of you when he wants to. He suffers no consequences for it, so why would he stop?
Also whar your daughter saw today is not something any young girl should be growing up seeing.
Watching her Daddy shame her Mum about her looks & presentation (totally normal & acceptable presentation) and repressing her, and leaving her at home, excluding her from what should have been a family outing ..... Probably cause he - at best - wants to appear single at his gym.
From a man with a history of cheating, as I said, I wouldn't trust him as far as I could throw him.