Good question, and I've given a lot of time to figure on it.
I've tried .22-250, and come back to the .223. For 600-700yd, the 223/5.56 chambering is decently accurate (I've fired it at 600 in national level F T/R) under match conditions (figure time restraints and some additional unspecified stresses), and it's just a very nice chambering.
The .22-250 is definitely more accurate out further, but it's another one of those magnum wannabees. The barrel heating is very hard to deal wit, and make it unsuitable for a competitive shooting pace/cadence. Accuracy goes to hell about halfway through the course of fire and it takes so long to cool off that the only really accurate rounds will be the first ten or so in the first stage. Period.
If you want to shoot woodchucks out to 400-500yd, there is no better chambering than the 22-250.
Despite all the options, the .223 turns out to be my goto .223 bore chambering. It's not the ultimate world beater, but in the end, it's perfectly suited for 600yd, and maybe out to 800yd. Folks go to some extraordinary lengths to make it work a lot farther than that, but it's a stretch, and it can get expensive quick.
If you want/need to do more, further; go with a 6.5. Mine has been the 260 since around 2001.
I have five 223 chambered AR's and two bolt guns for it. One of the bolt rifles is a heavy 24", and the other is an 18.5"
Mossberg 223 MVP Predator that may well be the most perfectly pleasurable carry rifle I have yet to own. I put
this scope on it, and along with an LR600 LRF, it's impeccable. If I ever have to go into survival mode, it will be my carry rifle.
Greg