I do agree that it is not the dog's fault. But I also think that it is not just one factor. He is old. As animals, especially small dogs, get older they get more aggressive. You would (and likely will), too. Does he have arthritis? Also, how long has he been so inactive? You made the decision as a pet owner to start a family. When you did that you decided that you would sacrifice time to spend with the kids and time to spend giving the dog the exercise he needs. And if he has those health problems, my god, OF COURSE HE IS SNIPPY. He hurts! I would look more into finding a way to get him the care he needs before putting him down.
You keep saying that you don't think there is anyone who would take him on as a pet, but have you actually looked? Have you called shelters and rescue groups and contacted the people who could help your little dog?
Of course you and your children come first. No one would ever suggest otherwise, but that does not mean that an animal can just be pushed aside because he is suddenly bumped down on the priority list. He has to make it in there somewhere. You are a stay at home mother, you can make room in your schedule for this particular member of the family.
You could even look at transitioning him to be an outside dog if you have the space. Dogs switch surprisingly well if you are consistent and supportive from indoor to outdoor. That way you minimize the risk of an incident with the kids. With this solution you do have to take time out to get him the attention he needs. Take the kids outside to play a couple of times to day.
I would also suggest taking time out to really work with him. Read up on if there is anything you can do to help make him feel better.
Try these: http://appliance.justanswer.com/JA/ASP_ASK/FID_12/K_794/P...
http://askapetpro.blogspot.com/2007/10/chihuahua-with-ele...
http://ezinearticles.com/?Natural-Remedies-for-Pets&i...
http://ezinearticles.com/?A-Study-on-Canine-Distemper-And...
http://www.ccrt.net/PDFs/WonderfulWorldofChihuahuas.pdf
http://www.earthclinic.com/Pets/bladder_infection.html
Please keep in mind that I am not judging you or reprimanding you. I know you want to do what is right. But honestly, if you really felt this were the right thing to do, then would you have asked on here? Probably not. You know what you want and what to do. I hope that this information helps and that everyone involved comes out of this well.
Honestly, if all else fails and/or you don't want to put the work into getting him the help/care he needs OR to find him a home, then yes. Put him to sleep. He isn't happy there, is in pain, and is not getting properly cared for. No one wants that.
Good luck. Feel free to email me.