I think you answered it. Since you’re already a reloader... get AXSR in 300 Norma now, add 338 barrel anytime for when you want to shoot factory for <$4 / rd.
And 6.5 barrel later...if you think you need to. Personally I’d add that to your AX308 instead.
Edit: Recommendation is assuming your AX308 is post-14. If not, 6.5 barrel for AXSR makes sense.
My ax308 is a post '14. So it has the quick change barrels. I have built a custom in 6.5cm, as well as bought a hunting rifle in 6.5cm for my son, so I'm committed on getting setup in that caliber already. I have a second bolt that I was planning on doing 223 conversion, so if the 264prc is a option I might go that way.
I don't see a 300NM available now is the problem. So my options would be get a 338Lapua, 338Norma or 300win.