Its entirely up to you and your abilities.
The question is, how comfortable will you be on a SMALL boat for a LONG time?
With the right weather window, I would think yes it could be done.
The River trip down and the ICW you should be fine.
Doing the keys and than crossing the stream to the Bahamas might be a differnet story.
BTW, most take the route through the Tenn-Tom waterway and don't follow the Mississippi all the way. You will take the Mississippi only as far South as Southern Illiniois than take the Ohio and later end up on the Tenn-Tom.
This water way drops you into Mobile Bay just West of the Florida Panhandle.
Courtney is My Hero
If a man is to be obsessed by something, I suppose a boat is as good as anything, perhaps a bit better than most - E.B. White