As someone who once carried a Walther PPK/S-22 for just under a year when I first started carrying a pistol for self-defense, I can tell you from experience that when folks advise against carrying a .22 rimfire pistol for self-defense, they aren't all just being caliber-snobs who insist that anything under 9mm is insufficient for the task.
Unfortunately, rimfire pistols are just plain finnicky because they're chambered in a rimmed rifle cartridge with an outside the case lubricated bullet and the very way that their primers are seated inside the case makes them distinctively unreliable.
My Walther PPK/S-22 feeds 40gr CCI Mini Mag FMJs with 100% reliability, but other brands or types of bullets have given me trouble, particularly Remington Golden Bullet, which I've had not only failures to fire and failures to eject will, but I've also had one stick in the chamber.
So yeah, I strongly suggest that you either upgrade to a centerfire cartridge of some kind, or at least switch to a revolver since that will eliminate the issue of ammo cycling which is present in so many .22LR pistols.
Otherwise, just keep experimenting with different types of ammo until you find one that your P22 likes, and stick with round-nose solid bullets.