Unless you charter a plane or boat, you will not find any commercial options for that route. Is there a reason you are stopping in Bimini on your way to Eleuthera? If not, then it would be far easier going direct from Florida to Eleuthera via commercial aircraft.
I'm guessing the issue with the commerical flights is bringing the 3 dogs with you. But if they are small, usualy airlines will allow pets on board for a fee. You will, also need to have importation papers as well as a permit from the Bahamas Department on Agriculture in order to briing them into the country, but if you have a house there, I suspect you already know about these requirements.
If the airlines don't work for you, then your only option will be charter plane or boat, which can get rather expensive. Getting from Bimini to Eleuthera presents its own set of issues -- largely a "you can't get there from here" sitiuation unless transiting through Nassau, which is why I asked if you needed the Bimini portion of the trip.
If you can get rid of the Bimini stopover, your best option will be to fly commerical (or via the Caravan from Fort Lauderdale, which does allow animals) from Florida to Nassau, then take the Bahamas Fast Ferry to Eleuthera. You could just opt to have the small pallet shipped separately and meet it in Eleuthera, which might present one less logistical challenge.