By car in August Red? Paris is pretty dead in August and a lot is closed (restaurants, shops, etc.) so tbh I wouldn't bother and look at other places depending on your direction, Versailles, Châtres, Orleans, Compiègne that would make a better day trip stop and will likely have nice hotels with pools that aren't extortionate. Le Havre has an amazing architecturally designed pool by Jean Nouvel but I assume it's not in your direction. Tours also nice but probably too far.
If you're dead set on Paris, I'd probably not recommend most Paris city centre hotels with a pool (unless you're looking at something like a Mandarin Oriental, Intercontinental or Hyatt in the Champs Elysees area but prices are probably crazy money range) tend to be tiny, very limited hours or by reservation only and not altogether child friendly or just randomly closed and this isn't usually apparent until you get there. It's also not great driving into Paris either so it's just a lot of added stress.
The La Defense area is not the most exciting but it's an easy train ride into Paris centre with the Rer A and there's a lot of business hotels there some of which surely will have a pool that's open most hours. Otherwise around Étoile is similar as a Congress hot spot that'll have a lot of business hotels with a lot of vacancies in summer.
Versailles is a longer trip into Paris but could also work (I think it's a half an hour ish by train but it doesn't go very frequently). Meaux is also a nice town and near Disney with a lot of hotels that would tick boxes around Serris but a longer schlep into Paris (45-60 mins each way), so I'd just go for Meaux and maybe some of the activities around Disney (food and shopping).
Can someone DM me the London dates too, I think when you were first planning them the dates didn't work for me but I think a different day was decided on later.