If I was to choose a sot for strictly DYI, it would not be GBI! If you have youor own flats boat and someone to share the poling, it would be ok.
I started thinking about using your kayak. You could get all over the north side, but wading in the marl would not be any fun all day. The fish on the south or beach side are not frequent enough. So unless you are going to fish from the kayak rather than just using it for transportation, the productive fishing on the north is still not going to be readily available.
I have fished on Great Exuma and wading is excellent. On Great Abaco Cherokee sound has a good reputation (as above), but I think you need a boat and guide for the Marls and I would want to go over to Mores Island. You could also rent a cottage in several places and some come with a small boat so you could ge to fishable beaches yourself.
I think for DIY, from what I have read Eleuthra is excellent and probably the one I would choose first. Second would be Long Island (I have a friend I would pump for local knowledge) and third would be Cat Island but the latter would require a lot of info gathering.