How about under $500. Stoeger Bullshark in 22 would be my first choice but if you don't like bullpups you could get a Ranger or Scout. They are all built by SPA. All should give you low 30s fpe at the muzzle and about 20 fpe at 50 yards. That is plenty. My little Prod is less than 20 at the muzzle and kills them just fine at 30 yards or less. They are relatively simple well built guns without many frills. Regulator can be adjusted but it's inside so it requires disassembly. But they normally come pretty much max'd out. So just adjust the trigger (simple but the stock has to come off), test pellets, set the hammer spring to a little under the peak, and go kill some squirrels. One of mine likes 18 grain, the other likes 21 grain. AEA have shot best this year but H&N and JSB will work. Both of mine have given me a 200 on the 30 yard challenge this year. They are very accurate, competitive with much more expensive airguns. Their simple design lets them be built to a lower price point without use of plastic or other questionable materials for any important part. If you really want a wood stock the Diana guns that are $600 or so are also built by SPA, have wood stocks, and I think the current ones have a gauge on the regulated chamber. Without externally adjustable regulators I don't think that gauge is worth much but others would disagree.
A less expensive airgun with a wood stock that will still get it done the SPA P15. It is the predecessor of the P35 which is the predecessor of the Bullshark. They are all similar but the P15 cocks in the rear. Krale has them for less than $400 but be careful about tarriffs.