My two cents -
If your .308 barrel shoots good groups and is in otherwise good shape, use it. The big struggle is to learn to read wind. You will be forced to do that with a .308. The .308 can be made to shoot well at that distance - ask any Palma shooter. Don't worry so much about super bullets. There are several good makers. Find a good bullet that groups in the rifle and learn to make it shoot.
The .223 is just going to be painful at much past 600 yards. Yes, it can be done, but it's tough to keep those bullets moving fast enough to stay supersonic and fight the wind.
Sorry if this offends anyone - I'm just passing along what I observed as I learned to shoot a .308 Palma gun.

Good luck.