I've traveled to Cuba via Mexico and Nassau over the years.
Going in and out of Nassau you just need to be aware that U.S. Customs and Immigration are right there to see you come and go.
It wasn't a big deal when I traveled out of there and should be less so now I would think. But it is something you should be aware of if you are traveling with out first obtaining one of the 12 travel authorizations required for legal travel to Cuba by a U.S. citizen. https://www.treasury.gov/resource-center/sanctions/Programs/Documents/cuba_faqs_new.pdf
And of course if you are not a U.S. citizen, don't sweat it!
I doubt there are any direct flight to Cuba out of Eleuthera. And a charter flight would not be cheap.
I liked flying Cubana. In the past it was Cuban rum and cigars available as soon as you took off. And yes you could smoke on the plane!