It seems that you have the legal right to seek medical attention as you want, but not the protection against a later fabricated claim or dismissal which they claim isn't related to it.
So we can reframe your problem a bit.
Your reframed problem seems to be, documenting that your work and performance are good, so that, if they later try to dismiss you, and you claim its retaliatory hence illegal, they can't stack the deck and fake a claim that you're a bad employee or have a poor record.
I'm going to read between the lines here. If you weren't a decent employee, you wouldn't be asking how to get help from a nasty manager (because you'd know he would hate you), or you'd be fired already (if he would fire someone for needing a doc for an hour, he'd surely fire someone for poor work). So I'm going to assume your work is actually good.
Now, this next bit may need to be done in some, specific legal way.
In my country I would simply ask to talk to him, say I want to check what he thinks of my work and if it's okay, does he see anything I could be doing better - and I'd either covertly record it, or covertly have an ongoing audio link (phone/chat audio call) to a reputable friend/ally, who could testify to what they heard and take notes.
In your state this could well be illegal, or need some research to find how to do it safely,or be proven another way. The idea works, but how you do it must reflect your own laws.
If later disciplined/fired, I would use it as evidence. I would justify the illicit recording as necessary, because the manager is known to be dishonest and lie about HR/employee matters. Note the neatness of this: if he is honest, it never comes to a court/lawyer/tribunal so the question of legality never arises. The only time it arises is in a case where the recording itself will prove him dishonest, where i tell the court i only took it because he is priorly known to be dishonest, as the recording proves in this case, thus retroactively justifying and confirming the need to have taken it so he can't lie to the court. So in my country I'd do it without a qualm. In yours i emphasise that you might need different evidence, or another route to prove you're a good employee. Perhaps a friend listening in, perhaps notes taken when you meet. Check.
Anyhow. You now have a legal right to a doc, and evidence cached that will tend to show you've been good and liked. If they retaliate for the doc, and claim you weren't fired in breach of your rights but only for being bad, you have your ace in the hole.
Principle - not keen on letting abusers get away with it because of exploiting the law to hide what they do. YMMV though.