Hi, Pit, and welcome to the forum. I'm glad you decided to post. There are a lot of folks here going through what you are. Be sure to look into the Caregiver's Sharing thread. Even though you aren't officially a caregiver, just having to take the kids to see their dad and your seeing him also, kind of puts you in that category.
If you've done some reading on here, you probably know that he undoubtedly has hepatic encephalopathy caused by high ammonia levels. You should ask at the NH if they are giving him Lactulose.
You also may know that a candidate for transplant would need 6 months of documented AA meetings or alcohol counseling before being eligible for evaluation. So you're probably right that that isn't going to happen.
When things start getting bad, you may want to call in hospice. Some nursing homes have the hospice nurses come visit and they are very good at evaluating for pain meds and making things generally as comfortable as possible near the end.
There is the possibility that he could get better if he abstains from alcohol.
Please remember to take care of yourself, too. I know it's a very stressful situation.
Hugs,
Connie