I am not sure why this is even a question...
Sail east through the Northwest Providence Channel between Grand Bahama and the Berry Islands. You'll have a fair bit of company from commercial
shipping that is doing the same thing you are. No need to stop and check in. I did this recently on a westbound delivery
Other posters have stated with confidence that the Bahamas allow anchoring under a Q-flag without check in. I do not know this to be a fact. I'd check it pretty carefully before relying on it.
The real issue with this route
is that you will be hard on the wind
the whole way. Nothing--legally--will require you to stop, but sailing close-hauled into the trades for a week gets pretty old (unless you are young and full of piss and vinegar!). In the winter you have the option of riding the back end of a cold front and getting most of the way there on 2 or 3 days of northwesterlies, but those getting really far apart this time of year.
Unfortunately, there is no magic route
that avoids this problem, unless you sail north of about 30N, but if you are in South Florida
that makes for a VERY long trip to the DR.
When we make our way east through the Bahamas we typically wait for favorable weather
rather than trying to make it in one go. It can take 2 or 3 weeks for the stars to align. That's cruising. It sounds like you are planning something more like delivery
mode. Do not expect it to be fun.