22LR? If I were forced- Solids and go for side to head shot, or WW power point to lungs.
22 Mag - much preferred w/FMJs to head
221,222, 223 w/reduced loads if noise is a factor, 52-55 w/PRECISE placement.
Personally, a 6BR can be downloaded w/but a FEW grains of powder and get say a 70gr at 2,000 - would do well w/head shots I believe, and reduced loads in a 243 should do the same.
Personally, I think a 223 w/the new tech bullets have a good enough track record to use w/confidence, but it sounds like you want minimal noise.
A 222 or 221 is very mild in report.
I would ONLY squeeze if using utmost care in shot placement and willingness to pass on anything but a perfect presentation and hold.
I took 4 rabbits w/head shots this year w/a LR and Mag at 50-75 yds...have to be VERY selective 'Timing' the shot so the animal does not snatch their head when you squeeze....as I wanted ONLY head shots for 0 meat damage.
Rule #1 - Destroy Vitals
Vitals die when hit with a 6.5