What "well found" sailing vessels can't make it to Bimini when it is blowing out of the North?
I've crossed the stream dozens of times. I've crossed without a lick of current and in over 8 knots of current.
As a general rule, use a factor of 2.5 knots as your average current between Fla and Bimini. That means, for a 10 hour trip, expect to get pushed north 25 miles and adjust your course accordingly. With GPS it's pretty easy to keep on top of your set, so that shouldn't be a problem.
As for weather, that's another matter entirely. On a 24' sail boat, sailing east in an easterly isn't going to be easy or pleasant. Much better in a SE wind, better still in a southerly, but any wind from the south will increase the speed of the stream. Anything north of east or west is just plain suicidal in a very small boat and a true norther has cost many, on well found sailing vessels of considerable size, their lives.
I really can't, in all good conscience, say go for it, since I do not know you or your vessel at all. I believe a 24' boat is pretty small for a winter time crossing of the stream, but if you pick the right weather, have a well found vessel and are a proficient mariner, well.... On the other hand, if you are on a schedule, you may well have to leave your boat over there until you can get another weather window to return which might be next year (or more) if you only have 1 week a year.