zebraYou could try playing music while shooting to drown out the sound. It's harder for people to complain about that type of noise during daylight hours. Complaints about noise aren't taken as seriously before 11pm (or there would be no workman on the road ever).
The fact is that they don't have that much power over what you do in your own home. They might be able to hear the air gun but the noise is not that loud in DB if measured and probably doesn't break any laws. lawnmowers are far louder.
If it's legal to use air guns in your neighborhood then, they can complain all they want but they have no right to stop you. If it's illegal then it's a different story.
You tried to be nice. Maybe it's time to let them know that your good nature has it's limits and you plan to exercise your right to enjoy your property doing legal activities whether they like it or not. They can can invest in triple glazing if outside noise bothers them so much.
Sometimes you have to just let people complain so they can see that they don't have the rights they thought.
When I lived in an apartment in Manhattan, the neighbors below would constantly complain that they could hear me walking on my floor. I wasn't going to give up my right to use the floor.... They were told by the building's management that there was nothing they could do but they still complained constantly. It only stopped when I started making complaints about them to show them how annoying it was. It turned out the building had much stricter rules about harassing your neighbors and putting threatening notes under their door etc.
Harassing your neighbors with constant complaints over nothing might be taken more seriously by the police than making a little noise during the day. If the police came, you could just say you were using power tools and tell them how crazy the neighbors are. They'll probably stop responding eventually.
If you give these people an inch, they'll take a mile! You're giving them too much power over you. They won't stop complaining even if you get an LDC. The noise of the pellet impact will be enough if they have an issue with you shooting.