@JellyWellyBoots lots of excellent advice given here.
You're obviously really clued up about what's what - is the block (to getting him into a care home/calling Social Services and saying 'That's it, now do something)- the guilt?
We all understand that.
Just two things:
Your words: You choose to have a child, so it's your job to look after them.
I didn't choose this.
Yes, yes, yes, yes, yes.
Secondly - to your unhelpful Aunt, and to him - 'Caring is a very hard job. That's why paid carers get days off. I don't get any days off, and I already have my own responsibilities. I'm not able to continue.
I can hardly believe I'm typing this, as I'm an animal lover/owner, but if he does need to stay, I would consider rehoming the dog. He'll go ballistic. But the poor thing will need to be rehomed anyway, and it might actually be helpful to start the process now - the dog could meet his new owners for visits, return to the security of your Dad - a gradual transition? I know someone who did this and it worked well.