Cooper Koch

I actually think Cooper's main competition, aside from his own co-star (Chavez, unless he gets category frauded into Supporting Actor) is the actor playing Aaron Hernandez on American Sports Story.
I agree with Chavez but I don't think the other guy will be a serious challenger at first (maybe he will be at the Emmys).
At least for the early Awards, like HFPA/Golden Globes, they use to vote for shows that are more global. As far as I know, the Aaron Hernandez show is a Only-US thing.
And let's not forget that Chavez could be nominated for Grotesquerie (also from FX) if the show ends up being popular.

Disclaimer (the AppleTV+ show) although from a smaller streaming has the billions from the Apple company, the most successful director of the last 10 years (Cuáron) and a big cast (probably the most expensive of this year - even considering HBO's House of the Dragon).

So, Cooper by now is at the top, and his stronger competition comes from Chavez and the guys from Disclaimer.