That is a really tough question, because there are rounds in all of the choices that will blow out a hole.
I personally would trade the DRT effect and some sewing for spending time looking for an animal.
(meaning that i can sew up a bullet hole faster than looking for a dead one from a round that is fur friendly)
I personally feel that your best options are 204 and 223. IMO
Plenty of guys i know that send in fur, say 204, and just as many say they stopped using those because it's not as fur recovery/friendly as they thought they would be compared to the 223.
It also comes down to distances you are shooting. if you don't shoot more than 300 very often these would both work.
I like my 223 and use 53gr vmax bullets (1:9 twist). I hate looking for them and most of the time don't have to go looking for a runner. Usually within 25yrds if they do run. I have found that even on Grey Fox, cats, and coyotes shot at more than 50yrds aiming mid body broadside works best. (use a shotgun inside that) I miss the ribs and the Vmax drops them. They don't blow out, and no sewing when fleshing. (the insides are all jello though) If shooting face on, i aim high neck so that the bullet doesn't hit between the 2 front legs, i try to hit an inch below the top of the back (the low neck/chest are is the one place that if you hit it seems to blow out often, and skinning is a pain when the front leg or armpit is blown out/off, easily doubling the skinning time)
I also do everything i can to not do head shots (unless i am using my 22lr pistol at 5' in the ear for a wounded not yet dead animal) Head shots with any AR caliber will make skinning and fleshing take longer. (your going to be scraping bone off the hide, and washing and cleaning the fur more, and skinning around the eye's is harder when they are popped out)
I like a can of sawdust, and trash compactor bags. The sawdust soaks up the blood in the bag, then the trash compactor bags don't leak, and a spray of flee and tick killer in there as you tie it closed for the ride back to the house takes care of the bugs. I also toss the bags in the freezer for a week min. to make sure the little bugs freeze and die. We have confirmed cases of bubonic plague here. I would rather not mess with that if i can.
Good luck and hope this helps.