I don't know that you'll be able to find a dive op that's going to allow you to take lobster while recreational diving with no experience. I don't lobster, so I could be incorrect.
I really like SubTropic, I've dived with them a few times. They have a good boat and a friendly staff. I've recommended them to many new divers and they've all been happy with the experience.
Think about going farther up the Keys though. Most of the dive sites around Key West are 60' or deeper.
Looe Key is fantastic, it's a shallow reef system a few miles offshore that has tons of juveniles, Goliath Grouper, and lobster. Your dives will be in 20-30 ft of water with great vis. There are several dive ops on Ramrod Key at MM27 that go out there. It's a short drive from Key West - about 30 minutes.
Farther up is Conch Republic Divers, widely considered one of the best dive ops in the Keys. I've had a great time diving with them and they have lots of experience with new divers. They're about an hour drive from Key West.
There are tons of ops in Key Largo and lots of fantastic sites, but that's quite a drive from Key West though.
You'll get many more replies if you post this to the Florida forum here on SB:
Florida Conch Divers