IMO the 243 gives you around an extra 50 yds point blank range over the 223, it also hit harder meaning fewer runners.
I have owned 223, 22-250, and 243, lots of people talk about zeroing 1" high at 100 yds, what i found out with this is most centre fire rifles go an extra 1/2" higher at 150 yds ( 1 1/2" ) if you then slightly pull the shot high off say quad sticks it can go through the foxes hair, so now i zero 1/4"-1/2" high at 100 yds with 22-250 & 243 and i can more or less head shoot hares out to 200 yds and body shoot foxes out to 250 yds point blank range.
The barrel life span on a 243 will be less than a 223 and it will cost around 20p per round.
I've often seen foxes jump through 6"/150mm wire fencing hence their bodies are less than this ------not as bigger kill zone as they look.
As for noise yes the 243 is louder but i think some moderators are better than others, i have a ASE SL5+ on my 22-250 and that is not noisy and is used for farms with housing estates around them.
223 22-250 or 243 all good for foxing but the latter two IMO are better for the above reasons.
Dave (warbucks)