You won't find .22 mags with more than 10 rounds... why? Because it is a rimmed cartridge, and doesn't stack well. You need a banana-shaped mag if'n you want more rounds.
I have some 12 round mags for my S&W 422, they are flush with the bottom of the grip.
Mags with larger capacity would have to protrude from the bottom of the pistol.
If you don't mind a used pistol the S&W 422 or 2206 would be an excellent choice for an accurate 22 pistol, I've owned both.
As for the 22 conversion kits, I'm not that impressed with them.
I would much rather have to seperate guns.