L.,
Speaking from and elderly persons standpoint. No, I do not believe you should let the neighbors know you are trapping them.
Here's a story, when my mother lived we us and we lived outside of town, Her and I started feeding the wild cats, it started with a pregnant female. She had 6 kittens and we were still feeding the cats. By the time we moved from there we had 57 cats that we were feeding everyday.
We gave them to the farmers in the area, because they all needed mousers. The cats would keep the mice out of their hay and there feed in the barns.
So ask your local farmers if they need some mousers?
Cats are great when you are older, they give us comfort because they cuddle.
I am a cat lover, we have two cats, they stay in the house and do not roam.
The health issue you have a legitimate, so do what you need to do for the health of your family.
If you are close to your neighbors, then discuss it with them, if not then just do what is right for your family.
Even if others in your neighborhood have cats that they allow to roam free at night or during the day. If they get trapped and sent to the Animal Shelter, well, they have no business allowing their pets to roam free in the first place.
You need to protect you environment for you family and that's it period.
Good luck
M.