Just my opinion,and I've never owned a 6mm.
These days,building rifles on milsurp Mausers is far less practical than it was,but there are still commercial Mausers with conventional Mauser magazine setups.
The 6mm is a 7x57 Mauser case necked down.Its perfect for a Mauser action,or any action that left the factory built for the Mauser family brass,including the 30-06 family.
If that is the sort of receiver I was working with,I'd go 6mm.
The popular short actions were made to work with 308 family brass.That includes the 243. If I was builfing on a receiver made for 308 family brass,I'd go .243.
I'm lazy. Square holes,square pegs. Round holes,round pegs. Yes,a round hole can be broached out to a square hole. As long as there are round pegs,why bother?
IIRC,initially Remington intro'd the cartridge as the .244,with a slow twist for light bullets only.They were marketing varmint high velocity,and limited the bigger game versatility.Once again,IIRC,I might be wrong,but the cartridge was re-intro'd as the 6mm with a faster twist.
Just my opinion,there is not significant difference in performance. 243 is easier to find on the shelf,but 6mm is not rare.
Whichever you have will do the same job.