That should be fine. Even a 1.5″ drain can be good if the vent pipe is close, but I’d stick with 2″, at least to the trap.

It could be that your original drain problem is because the vent is too far away, so that gases from the soap suds back up and cannot escape other than back up through the drain.

I don’t think I’m talented enough to explain any better without writing a novel. Hopefully you get the idea.