I would look for a Bruno #4 or any Anschutz (CIL's are great). The new CZ 452 Varmint and Rem 541's are great. You might need to change the stock to work off bags better. No big deal as recoil is not a factor. Bed and tune that trigger as well as possible.
Buy a bunch of QUALITY match ammo. Expect to pay at least $5 a box and don't cringe if the rifle needs $15 per box to shoot its best. That is still three times cheaper then low priced match quality centerfire.
Even reloads cost around $1 per bang.
I would look right away at the scope you plan on owning long term. May as well enjoy the best glass you can afford. If really on a budget, the Elite 3200 10X is a superb scope for dirt cheap. Burris sig rings with inserts so you can adjust for the drop.
Shoot and shoot and shoot.
Some will consider it a joke to use a 22LR. Just ask them to shoot your rig. Odds are if they are use to high velocity, high BC cartridges, they will do very poorly with the 22LR at 100 or 200yds.
It is the toughest LR cartridge to use...
Jerry
PS black powder cast bullets would likely be worse.