Q: What is the gun used by the US Olympic shooters in small bore running boar competition?
A: Custom 10/22s with Leupold double reticle, double target knobs scopes.
Really interesting little rifles. The rifle itself could be duplicated, pretty easily x scope, using pieces and parts available.
The custom built scopes cost more than your total budget by multiples.
A used 10/22 can be tricked out by near anyone with minimal mechanical ability. There is 1001 mods for them, including match barrels, triggers, steel screw in barrel actions, walnut stocks, mags, etc. A 16" 22 barrel is your friend for hunting.
They do shoot regular high velocity ammo, like CCI Mini Mags well.