I think that cats are tough little critters, so are coyotes, but not like cats, so I would suggest not using FMJ bullets or you will loose a few when they run off.
I would rather miss completely, than know i hit it, and have it run off and can't find it, and would also rather sew a hole than not find it. (personal opinion)
I think that i would suggest the slower twist (1:14 or 1 :12) 22-250 of even 223. given that you want to shoot 500, then I would suggest the 22-250. I found the 40gr bullets like the 14 twist most, but every gun is different.
The idea is to use the 40gr vmax, i had very few blowouts, and that includes 50yrd fox. Slow them down a little, and go for accuracy and the tightest groups you can.
The other thing i would suggest is focuing on shooting and putting that round in the right place to avoid blowouts. I personally avoid headshots (makes skinning nasty and time consuming) I don't like upper chest shots cause I tend to get blowouts, and a bloody mess when skinning. I like the broadside mid body, behind the front leg shot, usually no blowouts, and not a bloody mess. That, and the Texas heartshot.
The thing that i found is that you don't have to shoot them like you would a deer, a heart or double lung shot is great.. But, using Vmax bullets makes it so that you don't really have to worry about that. Shooting 3 or 4 inches further back on the body, drops them on the spot since the bullets are exploding inside the body, and it's still a heart and lung shot 4 inches back from these organs. Don't worry about gut shot.. It's not like you eating it. lol