As you brought to my attention, the Bahamian government page Applying for a Visitor Visa suggests that British diplomatic missions will process Bahamian visa applications:
Deadline
Visa applications should be submitted at least two weeks before travel. Applicants applying for a visa at a British Embassy must submit the application at least four weeks prior to travel.
Related Fee(s)
The following fees apply for visas obtained at The Bahamas’ Embassies and Consulates. Prices may vary for Bahamian visas obtained at the British Embassies.
...
(emphasis added)
Also, at Entering the Bahamas:
The Bahamas is a major tourist destination, so we make every effort to keep visitors’ Immigration formalities to a minimum. However, entry requirements do vary from country to country, so nationals of non-Commonwealth countries should enquire at the nearest British Consulate or Bahamian Embassy for information on these.
(emphasis added)
If I were you, I would note the exact words the British high commission used when they "said they don’t accept applications for Bahamas visas." If you have it in writing, so much the better. Then send an e-mail message to the address found at the bottom of Applying for a Visitor Visa under "For more information please contact." The first address is for "The Bahamas Consulate Division" in Nassau. Explain that their website instructs you to apply at the British high commission, and that the British high commission said that they don't accept such applications, and ask them what to do.
Once you have the answer, please come back here to post it so others know what you found.
