I have always left my D300 and D700 in 1/320 FP mode whenever I wanted the FP option - and that is almost all the time.
I never understood the reason for 1/250 FP - never understood why I would want to shut out the option of that extra fractional stop. If the camera offered 1/1000 FP that would be better still!
It is not clear to me at the moment what difference, if any, occurs when a 1/320s speed is used when the camera is set to 1/250s FP. Presumably it shoots using much less efficient FP bursts when it doesn't have to do that, and could use the more efficient standard flash sync. I might have tested it years ago and I guess I should refresh my memory .
So I think the question should be "who doesn't use 1/320 FP" because I think a good reason would be hard to come by
I was not aware of (or recall noticing) any edge darkening issues - I'll have to test that some time.
When I am shooting wildlife with a long lens I usually keep it to 1/250s because I don't want to lose that stop, or some fraction of a stop at 1/320s even in 1/320s FP mode. I usually want all the light I can get, if only to minimize the recycle time. Any edge darkening would rarely be an issue with my birding since I am typically doing deep crops anyway.