OK then. Bimini is definitely the wrong way to go! TWO options:
1. Leave out of the West Palm Beach inlet about midnight on a good forecast and set a course for Memory Rock passage. You will arrive at Memory Rock in daylight and go onto the Bahama banks in plenty of water. Proceed on in daylight to Great Sale Cay and anchor mid-afternoon. Enjoy and get some sleep...there is NOTHING there...just a good anchorage. Next day, start early and proceed to Green Turtle Cay...anchor off the New Plymouth settlement and take the dinghy in to check in with Customs and immigration and pay your fees. Wait for DEAD HIGH tide and enter WhiteSound channel for a protected anchorage and nice food/drink etc.
2. If you are not comfortable with leaving W. Palm at night...get a start at daylight and proceed to cross to West End on Grand Bahama. You will need 8-10 hours for the passage in settled weather. You can pull into the West End customs dock to check in and take on fuel if needed and then use the Old Bahama Bay marina there for a slip for the evening. Check the tides for the next day and proceed the 8-10 miles north to the Memory Rock Channel and follow it to Great Sale Cay as in option #1. Do NOT attempt to use the Indian Cay channel which is closer to West End with your draft.
For either passage you can also dock and/or check in at Spanish Cay Marina instead of Green Turtle should time or tide or weather dictate. Get the EXPLORER chart books only for the Bahamas...they are the only reasonably accurate ones!!
Once you are done with the Abacos, you can exit out of the Little Harbor passage and head to Royal Island/Eleuthra to make your way down to the Exumas.
Note...before you head across...there is a great anchorage at the north end of Lake Worth just off the ICW. You may have to wait for a week or more for the right weather to cross the stream and this is a good place to do it since you can take your dink to shore and shopping and West Marine are all close at hand.