This is REALLY good advice. OP I don't envy you. This is probably going to be one of the hardest things you ever do. We unfortunately had to make this decision when my eldest was a toddler. Our dog had been to tons of trainers, and was just a tiny alpha dog, he peed on a couple of the trainers. He was okay with me, but he sprayed all my husband and son's belongings imediatly when he came inside and constantly snapped at them because he was jelous they got any of my attention. It had gotten to the point we couldn't let him in the house unless he was in his crate

I remember Bear was sitting on me and my son came over to hand me my sippy cup and he turned and tried to bite him. I still miss him, but honestly he was better off with a single owner who had nothing better to do than stare at him. I NEVER imagined I'd be in that situation and felt like everyone condeming you, until it happened and it became a real possibility he was going to mame my son.
You know what, dogs do kill kids. It's not all that rare. Then it's guarenteed death for the dog. Rehoming it is right. Currently the dog is only threatening, if it bites it'll be game over for it. And sorry, I love my pets, but my kids come way way ahead of them. They aren't even close to being equal. I'd die for my children, I wouldn't die for my dog.
And all I have to say about the nursing home scenario is if you haven't had to put any of your relatives in a home you are VERY fortunate. Sometimes people can't stay at home. Sometimes you can't keep them safe unless there are multiple people watching them. We had a situation with my mom where she had enough money that she hired her own help, and kicked them out often. She sat in a chair coated in urine after she broke her foot because she wouldn't let anyone in to help her to the bathroom. She refused to bath so she got the nastiest full body rash I have ever seen. If we tried to help by bringing healthy food she'd refuse to eat it, then order chips and antacid from the grocery stores that delivered. She hoarded paper and smoked around it while losing control of her hand which led to her dropping cigarettes constantly, it was a miracle she didn't burn down her house. She ordered $30,000 of jewelry over the course of six months from the shopping channel. She refused to have anyone help her transfer into her bed, and there were weeks where we had to pick her up off the floor at least once a day. If we tried to intervene at all she kicked us out of her home because it meant she couldn't do whatever she wanted. I couldn't work from the age of 21 until I was 32 because I was constantly over there attempting damage control, and she was abusive the entire time. She messed up her drugs to the point that it almost killed her, fortunately she ran out of ativan and went into withdrawl which allowed the authorities to come in and declare her incompetant. We tried desperately to keep her into her home, at a certain point it was very very obvious it wasn't in her best interest but only to aleve our guilt when really we didn't do this to her, it's just life and it was way way beyond what was feasible to deal with outside of a nursing home. And since she's been in the home she is much healther, I have no doubt she'd be dead if she didn't end up in the nursing home.
Sorry about the off topic, but this drives me nuts when people refer to it like you are "dumping" your relative in the nursing home. If you can't understand how it could be necessary don't judge, think "there for the grace of God go I" and be thankful. So maybe it isn't so off topic, because unless you've had a pet that is a real risk to your children you can't get it.