Welcome to Traditional Muzzleloading. I hope that
@rich pierce spots this thread and jumps in to tell us about what rifles he has seen at the Primitive Biathlons in which he has participated.
What you need is an accurate on target rifle that is relatively light in weight with a large enough caliber that will be easy to load while getting exhausted trampling the course on snowshoes. The alternative choice is a small caliber, squirrel rifle using a loading block to ease the loading process. I'm not sure what rifles are allowed by the competition rules. To that end, as a new to primitive biathlons, I would look for a used Lyman Great Plains rifle in 54 caliber or a CVA Big Bore Mountain rifle in 54 caliber. Since the scoring is done by number of hits, a 36 caliber Traditions squirrel rifle would work too. I would avoid any of the smooth bored trade guns as the accuracy would not be competitive. The Traditions might well be acceptable with a good lock tuning. But I sure wouldn't want to overlook the used rifle market.
I found the website for the Lamoille Valley Fish and Game Club's Primitive Biathlon. Pretty much any type of Traditional muzzleloader (rifled and smooth bored) and some non-traditional categories can compete.
https://www.lvfgc.com/muzzle
So, get the muzzleloader you want and get ready to compete next year.