OP you need to phone
a Bahamanian Embassy and ask this question, they probably wont be able to tell you immediately.
But they'll likely tell you that you're means you wont qualify for an automatic a US citizen 30 day visa, and so you will need to apply for a visa and after a period of time it will either be granted or not.
No point asking here.
Also the decision to grant a visa or not will depend on the nature of the conviction and penalty imposed, especially if there was a prison (or equivalent like home stay) sentence. They'll require full disclosure. That's true for virtually every country in the world. They all have two doors at the border, a door for good people and another for potential bad guys.
That's not to say you wont be granted a visa, but you'll be pointed to the bad guy door obviously where there is a much closer analysis of your intentions during the stay. Clean slate wiping of convictions usually doesn't apply to visa applications, in the same way it doesn't if you wanted to get a security
clearance to work as say an FBI agent. And lying in a visa application is never a good idea as I'm sure you know.
in the same way as people with two passports (dual citizenship) might use the passport of the country where they have a clean record to apply. But there is considerable co-operation between Governments to stop bad people crossing borders. So Immigration officials usually know the answers to their questions before they ask the person concerned. And if the applicant lies in their answer they just kick them to touch without wasting more time or worse, lock you up.
As you'll already know, foreigners wanting to enter the USA just for a visitor's visa with a criminal conviction means the bad door. So a simple test you can do might be this: imagine you were from another country trying to enter the USA. Would the USA Govt give you a visa with your record?
The Bahamas, like many countries, provide automatic visas to citizens of various countries including USA. But in the fine print automatic visas are restricted for people with clean records (plus all manner of other exceptions).
But to save you a little time here are the requirements from the Bahamanian Govt's web site:
Contact the nearest Bahamas Consulate or Embassy for required supporting documents to be submitted with application form supporting documents vary from Mission to Mission.
Valid Passport - All visa applicants must be in possession of a valid passport, with more than 6 months remaining, and a return ticket.
2 passport photos on white background
A job letter/business licence
An original bank reference letter/statement
Flight itinerary/ticket copy
Copy of conference/invitation letter
Copies of previously held visas
Copy of Children’s Birth Certificate to prove parental relations
Copy of a valid work permit (for Companies applying for a foreigner to take up employment
in The Bahamas)
All foreign marriage, birth, and death certificate submitted in support of a visa application must be legalised/Apostilled in the country of origin before submission
Valid US visa (for Foreigners who wish to visit The Bahamas via cruise ship
from the USA)