I've used some 110s and factory 130's in the .308 and they are really destructive at full velocity. With reduced loads they start acting more like a pistol or 30M1 carbine bullet. Between the 2 I would go with the .22-250 and near full velocity loads. More chance of dropping the deer in it's tracks. The 223, 243, 260, & 7mm/08 are good choices too. Of these only the .223 would be noticeably quieter when wearing ear protection, which should be mandatory.

"When you disarm the people, you commence to offend them and show that you distrust them either through cowardice or lack of confidence, and both of these opinions generate hatred." Niccolo Machiavelli