Ballistic putty is like modeling clay. It is a semi-soft material that I have used to stop pellets from my airgun in my basement. I am sure that it would both slow the .22 bullet enough to lessen the impact on the rear of the trap, and it would bond to the back of the trap, also helping to deaden the sound by increasing mass. When the putty is full of bullets, you simplyremove them and reshape the substance and put it back in the trap. It is also very cheap. Iwould bet modeling clay works just as well.
Hope this helps.