Hurricane season runs from June through November with the peak storm activity during September, see the chart below:
Hurricanes and Storms
However the tropical Atlantic and Caribbean is a huge region, so just because there are storms in one area doesn't mean that the entire region will be affected; but the probability of storms does increase during September.
When it is not stormy the weather is often calm and gorgeous and the water is warm. I wouldn't hesitate to take a trip in September, but I would buy travel insurance just in case.
We dived with Blackbeard's years ago, our first liveaboard, and I have no complaints. We got a good experience for a bargain price, but the lack of privacy isn't limited just to bunk beds in noisy dormitories with snoring room-mates. There is only one bathroom onboard, adjacent to the galley, and you have to take salt-water showers and then briefly rinse off with fresh water. On our cruise they ran out of fresh water after a couple of days and we could only take salt-water showers. You can also get a fresh-water rinse with the condensate hose on the dive deck. We had fun and I'm glad we did it, but I wouldn't go back. There are lots of threads about Blackbeards on Scuba Board if you search the forum.
I also got very seasick during the deep-water crossing from Florida to the Bahamas, but now they depart from Nassau so that may not be a problem.
One nice thing about Blackbeards is that there is no single supplement, or need to share a cabin with a stranger, to get the cheaper rate offered for double-occupancy.
@10traveler10 has dived the Phoenix so perhaps he can give you more information and
@Brodydog was considering it, but I don't know if he ever went on a trip, see the link below:
Phoenix Liveaboard
We did a lot of diving in the Bahamas when we younger and had a lot of fun, but the diving wasn't as good as some of the other Caribbean locations we've dived IMO - but it is close and easy and you are likely to see sharks. Bahamas water is also quite cold (to me!) in the winter and spring, but it is nice in the autumn.
If you are also considering a land-based trip during September, I got an email just yesterday from the Clearly Cayman resorts that they still have some rooms available for their Throwback Special at the Cayman Brac Beach Reort in September, and the rate for single-occupancy is very reasonable if you are traveling alone, see the link below:
https://www.scubaboard.com/community/threads/throwback-special-at-cayman-brac-beach-resort-still-available-for-doubles-and-single-occupancy.579385/#post-8651743
I've been on diving vacations during bad weather, both on liveaboards and on shore, and I'd rather be on land during rough situations!