It's not enough to have a man-stopping round loaded in your revolver. You also need to be able to hit something with it, and that generally requires practicing with your carry ammo. If shooting the 360 with hot Magnums really hurts your hand, affects your accuracy, and the recoil makes quick follow-up shots impossible, then you and the gun aren't going to be very effective at crunch time.

There's a reason S&W sells so many 642s and 442s - because .38sp+P from a 15-oz alloy revolver is snappy enough for most people. Use hotter ammo in a lighter gun, and the snap factor goes up quickly.

J-frame Magnum rounds are best delivered from a stainless steel 3" with full underlug. Even there they can be memorable.