Ensign Garrovick from "Obsession". I could see him being a Captain by 2293 and in command of the Enterprise-B. He'd be in his late-40s or early-50s by this point. The perfect age to be a seasoned Captain or at least a seasoned officer finally promoted to Captain. Except I wouldn't want to do that to him and have him of all people carrying the burden of Kirk dying under his watch.
So, who would I have if it were someone from TOS? Lieutenant DaSalle would've been a good choice if it weren't for the actor's age. Michael Barrier was born in 1933. That wouldn't have worked. He would've been in his early-60s, assuming they stuck with the same actor.
No one else is jumping out at me from TOS itself. I'm not reading other posts before I type this, because that would be cheating.
It would have to be someone from the TOS Movies. I prefer the idea that whoever is in command of the Enterprise-B is noticeably younger than Kirk. Saavik is the best choice, but I agree that 2293 seems a little too soon.
Weighing everything else, I'd still go with Saavik, despite it being a bit too soon. She could've ended up on the fast-track after TVH. Not something that would be out of line with her having graduated as a Lieutenant. Her first interaction with Kirk on-screen is TWOK, talking about no-win scenarios. Her last interaction with Kirk would be this version of GEN where they win by saving the El-Aurian refuges but end up losing Kirk. It would've been a bookend of sorts.
BUT
I'm glad it was a new character. I'm not glad it was Harriman. He should be way more experienced than he was. At minimum, he should've gained a lot of that experience as a First Officer. As it is, he seems more like an Ensign or Lieutenant who's in way over his head.