I'd expect Paris to fire off almost all that's remaining, leaving just enough to blow the ship into scrap--having a battleship captured is not anything anyone wants to happen. Even if it's a totally useless hulk towed into port, that would be a major propaganda coup.
As for butterflies--a BATTLESHIP, even if an old one, has been destroyed by bombs alone. That will have repercussions in the aircraft vs battleship debate. It might even result in the USA putting a little (not a lot!) more into carrier construction--same with Japan. The first Essex was laid down in April of '41. In May of 1940, all of the North Carolinas and South Dakotas are on the ways; none of the Missouris are. (Indiana had been laid down a few months previously; the otehr three SoDaks have at least a year of construction already done. The first 6 treaty battleships are certain to be completed, but will the Missouris be delayed to allocate carriers at a higher priority? There's limited numbers of slipways that can build such large ships...might a battleship sized slipway be used for an Essex or Yorktown class carrier or two? (Yorktown if the Essex design isn't ready yet...)
The Iowas have not been laid down yet. In this timeline, accelerate US Navy shipbuilding by a couple weeks/months compared to OTL. Iowa is probably getting laid down in the next week or two of the timeline.
As far as the USN reaction --- that is what you get when you have an old battleship operating close to shore, in restricted waters as a glorified monitor when there is no fighter support.... I don't think they'll learn much as Paris did not sink. She decided to join the choir invisible and beach herself to support the army instead of limping back to port. If the weather was good, she probably could have made it to Le Havre for temporary repairs.