Also, if it's just about diving and not the adventure of visiting the island, I think some of the Akumal dive shops do offer Cozumel dives. Looks like Akumal Dive Center may do it twice a week. Maybe the others do as well. But going to Cozumel and seeing a bit of the topside would probably be more fun. Most of the operations send boats out around 8 am--getting on one of those would mean a very early ferry! Dandy Don covered the Blue Angel 10 am boat--as I recall, it's a slightly larger boat and may have more people on it than the fast little boats most people take out in the mornings, but BA is a nice operation, and I'm looking forward to diving with them again in a couple months (you'd probably need to call or email them for prices/schedule--website seems to be in transition). Their office is at the Blue Angel Resort, not right downtown but close. It looks like Dive With Martin (shop farther south) does a boat around noon, and I'm thinking a lot of others do too. I haven't been out with DWM for a few years, but we always liked them.
There are also the "big tank" operators like Aldora and Living Underwater (we've been diving only with the latter) who can outfit you with big steel tanks--their dives tend to cost more, but it might be worth considering especially if you're using a lot of air, because you'll generally get longer dives.
Either way--if you have a choice, try to go out with a smallish group and an operator that will give you a decently long surface interval so the second dive isn't too short. After that, it's usually all good. There are scads of dive operators out there, and sometimes it's more a matter of fit rather than who is "best." You'll probably like whoever you end up going out with, but it's always good to get your questions answered first. And I'm sure you'll want to let them know that you're new to diving--they'll hopefully keep a close eye on you and pick a site appropriate for your experience.
We did our very first non-lake dives on Cozumel nearly 20 years ago--it was easy and awesome. Still is. Enjoy.