Same I have the iPhone 12 pro max I originally had the iPhone XS Max. Ever since I bought this phone almost 6 months ago it is had the issue where it lags really bad when I notification comes up. In my experience it makes no difference what game you’re playing and if it does then it has been every single game that I have played on this phone. Which is quite a few games but at this point I can’t take it anymore I have gotten into multiplayer games more recently and it is completely destroying the game for me!! I tried the picture in picture fix and it did not work at all for me. I am currently on iOS 15 as well and it has done this on each iOS version since I have owned my phone!