Oldschool280,
You have a what is essentially a Long Action in the M98 pattern. This action will hold any cartridge up to and including the .30-06 family without too much adjustment.
Like any form of competition, F-Class is a learning curve. No one is born with all the experience, skill and knowledge to make a run at the top spot immediately. Most of us start out cautiously, testing the water before we jump in all the way.
The 6mm Rem. cartridge is a good performer with the 105 grain Berger Hybrids. Why not check the twist of the 6mm Rem. barrel already in place to see if it is suitable for shooting the longer VLD bullets which require a fast twist. If it is, check the condition of that barrel to make sure you have enough life left in it to make it through a season of casual competition. If not, move on to Plan B.
If your FN is a magazine fed action then you can select any of the long action cartridges for the .30-06 family. A 6.5-06 standard or AI version will perform quite well for your introductory participation. You will get the similar performance to the 6.5-284 Shehane cartridge. Or, let your gunsmith chamber the new barrel for the 6.5-284 Shehane which will run nicely in the longer action.
The same situation can be had by stepping up to the .280 AI and shooting those 180 grain Berger VLDs. Be sure to check the twist rate! You may have to scale back a little to the 168 grain VLDs if you don't have the twist rate needed.
You can run those short action cartridges already suggested but you will need to have the Mauser magazine modified if you choose to shoot from the magazine. You could also use a temporary single shot follower for any of the cartridges suggested.
Be aware that as you grow in the sport, you will want to grow with the equipment. Custom actions with custom barrels and custom triggers bedded into a custom made stock are de rigueur for the tougher levels of competition. Add to this the expected cost of those high-end competition scopes and front rests! The U.S. F-class Open Team is shooting the .300 WSM or the 7mm WSM depending on the Captain this year. The Brits, Canadians and South Africans, all top winners and competitors, have chosen the 7mm RSAUM or the 7mm WSM cartridge for the horsepower and BC for wind drift. But don't think you have to start out with these cartridges or that equipment just yet. Work your way up to it and enjoy the process!
Regards.