Like others above, in the Bahamas all I've done is a Blackbeard's liveaboard cruise in the Exumas Cays (leaving from Nassau). We did as many as 5 dives a day, and it was pretty good diving -- though not as good as Cozumel. From everything I've read (a decent amount) liveaboard diving is better than land-based options in the Bahamas.
Though I'm glad we did that, I prefer both the diving and the top-side activities of Cozumel.
@loady will you be by yourself or have a buddy with you? Was not sure if that $3K budget was for 1 person or more?
One reason I ask is that Scuba Club Cozumel might be a good option given your parameters --
here's their latest special. However, most of their advertised rates seem based on double-occupancy, so you may need to contact them to find single-occupancy rates. SCC has boat dives and unlimited shore dives. Some people love it, and some people don't -- mainly because they use bigger boats with more people, rather than the 8-pack fast boats that many Cozumel dive ops use.
Another option is what I did on my last dive-focused trip: Dive with a good dive op (I've done about 10 trips with
Tres Pelicanos (3P)) and stay at
Cozumel Hotel & Resort (Wyndham) which was less than $100/night for the room including daily breakfast buffet. While Cozumel Hotel does have an on-site dive op (Dive Paradise) I prefer the smaller, fast 8-pack boats that 3P has, and I really like their people (from good communication while booking, to nice office people, and divemasters good at finding small critters, etc.). After looking at prices separately online, I ended up getting a hotel + diving package deal through 3P that was the best value for me. We went with the breakfast option and not all-inclusive, because there are so many good restaurants in Cozumel with reasonable prices. Cozumel Hotel does have shore diving available -- you just rent tanks & weights from their on-site dive op (believe it was $15).