Diving around Puerto Aventuras and Akumal, doubles would be a waste of time (unless you are doing it for training/practice). I've been diving in that area for several years and getting even to 90-95ft is not a daily thing. Most of the dives are 80ft or less (mostly less).
There are a few shops in Cozumel that advertise tech diving and others that routinely go to the more advanced sites. You're looking at about 2-2.5hrs door-to-door from PA (depending on traffic and timing with the ferry). It's about $20-25 each way for a taxi and $12 each way for the ferry.
One thing about doing day-trips to Cozumel though is that you'll be hard pressed to get anyone to take you to the most advanced and deepest dives there until they can dive with you for a day or two and assess your skills and comfort level (them with you and you with them). If you just go over for a day, don't expect to go to any northern sites or to places like Maracaibo deep, Punta Sur (Devil's Throat or Punta Sur Sur), Columbia Deep...