With a trailer sailor being a smallish boat you will want to leave from Miami. Dinner key or no name harbor are the usual jumping off spots.
Crossing the stream in a small boat is doable but not a trivial exercise. Make sure you have charts
for ALLthe Florida inlets up to and including St Augustine in case you need to turn back. Also do not rely on making Bimini. The stream can run at 4knts + so have contingency plans for other Bahamas land falls.
WAIT for a good forecast. I have sat in Miami for a month in October November waiting for one. At the time I was sailing a 38 foot ocean going steel ketch.
At another time and further North I got caught out in the stream when the wind went North. It is the ONLY time in 7 years sailing that boat I had water coming in green over the foredeck and the ONLY time I feared that I might get rolled over.
Lots of boat ramps. Don't know about long term secure storage of a car and trailer.
Your Bahamas fishing permit comes with your cruising permit. I am sure that the bahamian authorities will be OK if you arrive with a cooler full of fish.
Assuming you want a haulout then Marsh Harbor in the Abacos would be my preferred location. Be prepared for sticker shock.