I don’t have any good advice, just to let you know that I sympathize with your situation.
We live in a small rural town where there are no fences around houses and have dealt with dog issues since we have lived here for over 20 years.
Now, I like dogs,(I’m a cat person myself) but I’ve always believed that if you are going to have a dog in town, you have to be responsible and some people just aren’t.
When my kids were little, we had a neighbor who kept the dog (a rotweiler) tied up outside. Every once in awhile the dog got loose. I had a friendly enough relationship with her that I was able to talk to her directly. she dealt with the problem and moved away soon after.
We also had some dogs that were getting loose and pooping in our yard. Again we talked to the neighbor but we were already friendly and he liked us.
This neighbor put up an electric fence and keeps the dogs inside at night so all is good now.
Currently we are dealing with rental neighbors a few houses down whose dog barks and howls all night long.
I would not approach the house(these people are scary) so I contacted the city who advised us to contact the policed dept. I did so. The police visited them twice, problem seems to be solved for now.
In your case, I like the letter idea(but I would NOT say you are going to shoot the dogs.)
If that does nothing, then I like the old Colonel’s idea.
I mean, you have already found the dog in YOUR yard!
I agree you need to protect your children.
As a matter of fact, if it were me, I think I would go straight to do what the old Colonel suggested.
a fence wouldn’t hurt, either, if you can get one put up.