For sporting guns the NRA has the Light Rifle program. Any safe .22 rimfire with minimum 2 lb. trigger and maximum 8 1/2 lb. weight. Time is usually 1 minute/shot. It could be 20,40,120shots depending on the match program. And could also be shot from various positions.[Standing, kneeling, prone].
The CMP has a similar program but theirs includes rapid fire events. 5 shots in 30 seconds for manually operated guns[ bolt action lever, slide action] and 5 shots in 25 seconds for semi-autos.
For either of these assembling a 10/22 or bolt gun would be a interesting adventure. zoomies' ideas are good ones. This type set up could also be used in Rimfire Silhouette.
The NRA also has the traditional precision prone and position matches. If you're building a gun for this type of shooting be prepared to write a lot of checks.
Remember that shooting competitions are really social events. It's not a blood sport. People who enjoy shooting getting together to have fun. Shoot whatever you have.
You project looks interesting. I might do something like that myself.