Yes, I think Obito would beat him. I simply don't see Gai being fast enough to land a hit on Obito in order to bypass Kamui since Obito was confident enough he is fast enough to Kamui someone like Minato if he were to start Kamuing right away and I see Gai entering the Gates which would basically result in Obito using Kamui until Gai drops the Gates and then Obito can kill him.
The Uchiha Kaenjin, on the other hand, would stop all of Gai's attacks if needed and result in Gai's fist getting burned. There is also Obito's Genjutsu and as far as I remember, Gai looked into Obito's eyes multiple times.
Not to mention that if they were to engage in CQC, I see Obito eventually either warping Gai or killing him off with Sashiki. Lets not forget that Obito's chains even caught someone like Minato though.
It's a combination of speed and unpredictable tactics. As we've seen in the fight between Kimimaro and Lee. Kimimaro and Lee were matched in speed or Kimimaro was faster (this.most likely the case), yet the moment Lee became more technical all the speed in to world couldnt help Kimimaro amd Lee was able.to land a blow.
Also, sharingan readings ones movements is one thing, as the body has to be able to reaxt fast enough to avoid the contact. Lee expressed this against Sasuke, as I'm sure you know.
Overall, with Gai already having excellent speed and the ability to display unpredictable movements continuously (beast mode nonstop) Obito will be forced to read multiple moves and try to have each portion of his body react, while trying to materialize to counter. In this situation Gai has a slight advantage.
Minato used a technical maneuver to hit Obito. Obito was using straightforward attacks, to counter Minato became technical, something Hai displays continuously throughout his fightz.
Kaeijin wouldn't be smart to use seeing as ot wouldn't help Obito is the situation. Obito will be shielded, but would need to leave in order to make contact with Gai. Also, he wouldn't have the time to perform the jutsu
Minato was caught by chains, but realized this before fully being wrapped tight. Gai's strength level is.on a different level compared to Minato, gai would use the chains against obtio seeing as he needs to HOLD the chain, attacking his hands or pulling them forwards counters that. also, GAI could simply jump oit of the loop