Up to 22" barrel (the longest 22 I own) CCI SV remains below 1080 fps. In shorter barrels3-4" it is louder due to unburnt powder at muzzle velocities as low as 850 fps.
My recommendation for the original question is M&P 15-22 pistol (8" barrel) or carbine (16" barrel), or 10/22. I find the 15-22 more reliable without tuning, and the magazines are easier to load. Bolt gun would be Ruger American, or CZ457 if you can pick up the American model on the cheap.
Get a cheap silencer, as they all perform pretty much the same. What the extra money gets you is exotic materials and reduced first round pop, which is not that big a deal for most people.
Optic- get a Primary Arms Classic red dot, or an economy rimfire scope.
Here's Dubi's 15-22p with $200 HTA silencer. This is a registered SBR. The Aimpoint H2 doesn't work any better than the PA Classic red dot

And my CZ457 which slightly exceeds the $600 budget