I am not worried about my back yard, but I have several large oaks and pecans in the front yard, which are just as infested.
Thinking about a pistol for concealment purposes; just get under the tree and shot about 10 times.
Honestly, I have not run into a pellet pistol that works well, and I've had many. The RWS break-open kind might be worth a try, though.
As for not worrying about being seen, good luck. Hopefully your town is more sensible than most. Here in Northern Virginia, Barney Fife treats an air gun discharge just like he does the discharge of any other kind of gun. When our pastor moved here from a rural area, his boys all loved to play with pellet guns. After a very unpleasant visit from the cops, which only had a happy ending because the boy involved was very young, they had to get rid of them.