Caution: I'm relying upon 10 year old memory - you may not wish to.
I expect Six Shillings to Beacon or Ship Channel Cay (via Flemming Channel) is the route CaptainRic described as “no problem /w 5' draft”.
I planned (charted & discussed with “locals”) the reverse course (Allen’s Cay North to Six shillings then Spanish Wells) many years ago, but never made the trip. At that time, I had reliable advice
that my 5.5' draft was no problem.
Being only a short run (30 nm ?), speed shouldn’t be a concern.