I have not ever shot in a rifle competition but I have shot varmints (prairie dogs) with both of those cartridges and also steel to 700 (both) and 800 with the 6mm.
The 6mm will out do the .223 in the wind. My .223 is around 10.5 lbs and the 6mm is 12 ish. I can spot my own shots with the .223 easily. The 6mm I have been able to spot them at 6-800 yards but not as easy. I’m assuming you would be allowed a spotter so maybe not a huge factor for you.
The .223 is much cheaper to shoot. Looks like an 8 twist in that rifle so I would look at 77 Sierra or 77/80 Hornady match. Those bullets put it closer to the 6mm but a typical 55 SP will drift significantly more. If it’s for fun then don’t worry about any of that and go have fun.
Get the best optics possible. At 300 yards those critters look pretty small. Sounds like a great time!
