Yea if they don't recommend it i wouldn't do it. Although i did witness an old timer once put 3 rounds in a doe before she could take the next step. Folded her like a rag. But he had been shoot for 2 or 3 times the time i've been alive so i wouldn't dream of doing it. 243 would be the lowest i'd go. I have a friend that has a bobcat jumping up to grab a pheasant that he shot with a 30-06 on a quartering towards shot. Bullet impacted just above the front shoulder and ran down the length of the cat about an inch wide. Taxidermist fixed it up and if he hadn't told you you could never tell the cat was ever shot
I know it's been said, but like to reinterate....Do not use a .22 for deer hunting. It can be hard enough at times to put them down with a bow or gun. Also, I don't think a .410 should be used either. Basically the same thing.