I use a .223, and I have had no trouble downing coyotes at distances much further than 250 yards. It's just a matter of practice. Most of my encounters hunting coyotes; however, happen at well under 100 yards. It's only when I'm in the pickup or on the tractor and just happen to run across one out in the open where some distance is involved.

As far as fur damage, the slower velocity the round is, generally the less damage it will do. For factory rounds, I like Hornady's 50 grain V-Max bullets. Until I started reloading, I used those all the time. I found them to be very fur friendly because the bullets generally fragmented very quickly, leaving fragments all along the inside without a large enough core left to push through and create an exit wound. Most .223 barrels have fairly quick twists, which helps also.

I wouldn't worry about converting a .223 into an ackley improved. If I was going to do that, I'd just get a .22-250 and save myself the cost of the conversion.

Also take barrel length into consideration. My .223 has a 20" barrel. A .223 AI or a .22-250 isn't going to give me much of an advantage unless I get a longer barrel.