First, I think the OP sounds like a neighbor from hell. If my afternoon was disturbed by the sound of repeated gunfire 100 feet from my window I would be furious, and call the police. After the first call to the police is logged, any evidence of stray rounds becomes lawsuit fodder for reckless endangerment.
Second, a bullet trap is great, as long as you and everyone who ever uses it never misses. But if you are loading and unloading a gun in your yard, there is simply too great a chance that someday a bullet will go over the target and into a residential area. Not to mention ricochets. This is a lethal weapon with a long range, not lawn jarts.
Third, if you shoot toward the same ground for a reasonable period, there will be lead in the ground. Regardless of what you think about that, you and your real estate agent are legally obligated to disclose that fact. And if your agent hears about it from your neighbor, he STILL has to disclose it to perspective buyers. That's a stupid reason to fail to sell your home when you need to.
"Ensuring your backstop" means a 10 foot plus berm, long distance or other geographical barrier. Not a wood pile or bullet trap. Shooting in a suburb is rude and dangerous. Put up a badminton net or take up woodcarving - this is a dumb thing to do in the yard you describe.