The slap of a pellet hitting the actual target itself is probably hard to get rid off, well unless you change the material the target are made off.
But for regular cardboard targets i assume the only way to quiet that down is to have the target holder at the end of a pipe that is then fitted with sound dampning on the inside.
That way i assume the impact slap will be reduced some before it get out of the pipe ??
A pipe NOT terrible larger than your target papers, i dont think i 3 foot length of 2 foot diameter pipe will do much.
I am thinking the regular ???? is it 5X5 " papers, anyway that and then a pipe that is 15- 16 " diameter, where some are lost to sound damping on the inside.
This will probably necessitate a light source on the paper, but that should also be quite doable.
Of course that in turn will make it a little harder to change target, though if in consideration when setting the things up, it should be a minor nuisance.
On the other hand if the setup is totally legal ASO and it is just the worry of a neighbour that’s not friendly, and of course American police having to react as they must.
I would just leave a MSG with the police, so they know if there is a call about "gunfire" at so and so address, they can be relaxed in their drive and arrival, which is probably NOT what the neighbour want.
Also if neighbour is fully aware it is a pneumatic air rifle, and call in a firearm discharge complaint, shouldn’t you go after him for filing a wrong complaint. ( Swatting attempt maybe )
My favourite papers are those 5 dot ones, i think for shorter range with pistols, but i use them out to 75 M