Unfortunately he is mean as well as just doing what roosters do when they’re together. He attacks people, too. The alpha is one of the sweetest roosters I’ve ever had; he has never attacked anyone. My 5 year old niece can pick him up. Meanwhile, beta rooster will go after anyone and anything. If he was at least friendly with people, I’d consider rehoming him.
With my friendly alpha roo, it would work if only the beta had a more calm disposition/wasn’t so vicious all the time. He’s always trying to pick a fight with the alpha. Amazing how two roosters can come from the same eggs but have totally different temperaments.
I think your choices are:
a) divide the coop or make another coop
b) eat the mean beta rooster
c) let someone else eat the mean beta rooster
Personally, I can't see any good reason to keep him, given your description. But he's your rooster, so it's your decision.
If you are determined to keep him alive in a good home, then I think a separate coop space is the only reasonable solution.