Yes the .22M is more but not enough to change outcomes IMO. It's a big long shot that the difference is gonna have any additional effect. Never fear, that's why we have .32's and .38's if you're worried a .22lr isn't enough.
While I would have agreed with that statement a year or two ago, the testing I have seen on the defense specific .22 Mag rounds, demonstrated a monumental
improvement in terminal performance over .22 lr. They achieved low end .38 special performance.
Here's one of the rounds.
The test data that I saw had the round expanding to over .40 caliber, retaining nearly all of it's weight, with penetration of around 11 inches. Better than some .38's do, in a 7 shot, 10 ounce gun with virtually no recoil.