PRS is a totally different animal than high power, then DMR, than informal plinking...etc.
I am a huge fan of the .243 Winchester and use it for hunting, PRS and F-Class...granted in 3 different rifles. That said, I am not sure a 6mm is the best choice for a newer person exploring precision shooting in its various forms. While I have a small following of people who have bought RPRs in .243Win, it is a specialized load and rifle that make for a great marriage, and you are handloading it. I would be surprised if the 6mm bullets you have are decent for precision shooting. Made a 1007 yard hit on my first stage of my first PRS match...so yes, Kraig is right, the 6mms are plenty.
The 6.5s (6.5CM or .260Rem) you can go buy excellent factory ammo, but not quite as much hunting ammo. It is nice to be able to buy a box of factory fodder to make sure it is you jerking the trigger and not your load development.
Barrel twist is important, as is a decent trigger and solid stock.