A 6br would fit your bill for a not so common(outside of the F open and BR circles) cartridge. It uses a 308 family bolt face and the components are readily available for the hand loader. You would be hard pressed to find a milder recoiling more accurate cartridge easily shooting very small groups to 600 and beyond. There are many cartridges with a powder capacity between the 6 BR and the 243 designed from one of those two cases if you decide to go faster in the future. Reloading data for projectiles up to 115 grs is readily available.