First off @JustAllRoundShit . I really think you need to put yourself and your wishes and your sense of guilt to one side. Why am I talking about sense of guilt? Look at some of what you said, pulled together.
"It must be terrible for him to be so out of control and helpless, completely at the mercy of the carers. I would hate it as well."
"He keeps crying, begging me to let him go home. I just don't know what to do."
"... but I feel I owe it to him to try."
Can you see that all these statements are about you, and how you feel? Please don't let that cloud your judgement here, try to look at the situation as unemotionally as possible. I know that's difficult, but you must.
Also - it's not unknown for those who have had a stroke to have altered emotions afterwards, and to be unable to control their emotions as they once did. I wonder if this is in play here. You describe your dad as "crying, begging me" and I'd hazard a guess that you've never seen him like this before. And for that reason, you might be interpreting it as how deeply he's feeling it, how desperate it is. But it might not be the case.
(https://www.stroke.org.uk/effects-of-stroke/emotional-changes#Difficulty%20controlling%20your%20emotions%20(emotionalism))
"He's fully there cognitively but very apathetic in the day. Just wants to sleep all the time. When he is not sleeping he is very agitated, aggressive and very quick to anger with lots of shouting."
Is that more of less quick to anger than pre-stroke? If more, consider the emotional lability mentioned on that stroke website. If it's the same level as before - all the more reason not to inflict him on your mother. Apathy - is it apathy? It could be, emotionally from the stroke. Or is it just that he expects people to run around after him? You describe him as "very dominating", so it could be that too.
"He is currently in a rehab home kind of thing. He seems to be slightly improving there but he absolutely hates it and wants to go home."
It is still probably the best place for him. He will be getting therapy, physio to help him recover physical function as much as he can. I would emphasise to him as much as you can that he avails himself of the therapy as much as possible.
"I'd like to bring him home for a week to see if it can work and if it doesn't or if my mum says so then he'll go into a home but I feel I owe it to him to try. I just can't imagine leaving him in a home for the rest of his life (most likely) when he's so desperate to go home."
But your mum won't say so, will she? It's just you trying to shift the responsibility onto her shoulders, which you know are not strong enough to withstand the hectoring she will get from him. If you were going to 'bring him home' to YOUR home, that would be your choice. But you wouldn't be. You'd be burdening your mother with an out-and-out bully, and she's been conditioned by that bully over decades to accept his bullying. You speak of what you owe to him - what do you think you owe to her? Do you owe her a miserable last few years to her life? Really? You owe her to make the last few years of her life "getting yelled at all the time"? Is that really what you hink is a reasonable choice for you to make?
"What would you do?"
I would do the right thing by both of them. I would arrange for his continued rehabilitation at the facility he's currently at. It is the best-placed to recover him as much function as he can. Your mother can visit, and it is within her power to leave when his "very agitated, aggressive and very quick to anger" behaviuor gets too much for her.
If you owe your dad anything, you owe him not to be remembered as a tyrannical bullying arse.