Have done this several times as
captain from Bahamas.
The US
government has no issue with any citizen departing the US (we do not check out to leave----yay freedom!)
The Bahamian
government (who have been unfailingly friendly and polite even in the face of really stupid provocation by some clueless boat owners) only want to know if you are leaving. A picture of the cruising permit will easily suffice.
The "problem" is the airlines and this is a problem not unique to the Bahamas; it applies to literally all international
commercial flights.
If an airline lets you on and the immigration officials in whatever country you arrive in deny you entry, the airline becomes responsible for getting you back to wherever you left. If you have a one way ticket, they lose a paid seat. Therefore, it is the AIRLINE that has the concerns. You need to prove to them that you are not going to become a burden.
You can buy a full fare fully refundable one way ticket and cash it in when you get home. I have personally done this. But the airline agents are wise to this so it can be hard to pull off at the airport. Have had one friend denied a ticket at FLL because she stated her intentions to the agent. Buy it in advance.
Silver and the little Bahama airlines have seen one ways for crews a million times and they will accommodate you based on a
photo of a cruising permit and an
email. You can deal with all of this in advance with an agent in FLL on the phone. The big US carriers--like always--will depend on the agent you are dealing with. Some know, some don't. Obviously FLL sees more boat people than Kansas City, but you will still face teh luck of the draw.
Once you land in
Bimini the Bahamian officials (in my experience) will be courteous, helpful and efficient. They are great! Very different than the scowling bad attitude officials at MIA who 'welcome' us back home!
If you really want to stab yourself in the eye with your thumb, look into the return policy to the US, but this is mainly an issue for your
captain. If that is not done under the local boater option you still have the option of turning up at a US custom facility 24 hours later. (Opa Locka will be the least painful)
Go. Have fun. Bimini is great and the blue of the deep
water will not easily be forgotten!