Caveat: I haven't done this trip, but I have researched it from the standpoint of both personal sailing and a paid delivery (which didn't pan out). So, there's only "book learning" + several decades of sailing experience and lots of offshore miles in these comments -- no "local knowledge".
As you've surmised, the issue of how to get from Florida to Honduras is all about what to do with Cuba, or more precisely how to get around it -- East side or West side.
First off, allow me a brief Cuba diversion: If you want to bring your boat home again, don't go to Cuba. It's not worth it and your insurance company won't cover you if you go there in any event. If you want to go uninsured and don't mind having the USG make trouble for you....go ahead. But, even if you go to Cuba you still have the issue of how to get around it.
Go to the west, its obviously much shorter, but as "Gtod" has pointed out, you will face a significant contrary current (2+ kts) first in the Gulf Stream that sets east between Florida and Cuba and another current that sets more or less north between Cuba and Yucatan. In boats with a long water line and big engines that is not a big deal, but you will be in a small boat with limited fuel capacity. With a 30 ft boat you'll be doing extremely well if the engine (or sails) will drive you consisently at 5-6 knots through the water. With a 2+ knot current that means something less than 3-4 knots over the ground in favorable conditions. I don't have charts handy, but I'd guess that the distance from Key West (or Dry Tortugas) to coastal Mexico is 300 miles -- assume 100 hours under really good conditions. How many hours can you motor? My guess is that in a 30 ft boat you can't motor 100 hours. Which means you'll have to sail a significant portion of it, which means it may take more than 4 days...and as "gtod" indicates the you'll be sailing on a lee shore... and if you're single handing or short handed you're going to get really tired after 3 days.... in short, if I was going to do this in a 30 ft boat I wouldn't go west around Cuba. But, if you are successful in making the passage across to the Mexican coast, you then should have little trouble working your way south day sailing along the coast to the Rio Dulce.
Go east and you'll have a longer, but probably a more "user friendly" trip. You'll have to work your way up towards Key Biscayne and cross the Gulf Stream there. The passage from Sarasota to the Miami area will give you an opportunity to shake down the boat and improve your skills. Then playing the weather you'll move southeast through the Bahamas to the Turks and Caicos / Great Inagua. (There are several books on how this is done and lots of boats do it every year). From Great Inagua you jump off on one or more long offshore legs -- but the wind and current are basically with you all the way. Along the way you can stop in Jamaica and/or the Cayman Islands for water, fuel and provisions. Again, I don't have charts handy, but my guess is the trip from Great Inagua to Montego Bay is probably 350 nm...and from there to Rio Dulce is ~ 500 or so. Downwind with a small winddriven current behind you should easily do 100 nm / day.
From all I've read, Rio Dulce is probably a better place to aim for as there are places where you can leave your boat and you'll be much nearer to inland public transportation.
Only draw back of the long route is that it could take 6-8 weeks to get to Honduras, and you said you had about 3 months. That doesn't leave time for much more than a quick visit, before you sail home via the Yucatan Channel, which as "gtod" said should be pretty easy because you're now going with the wind and current.
Putting myself in your shoes (I too have a daughter who moved overseas for a time and so I know what fun it will be for you to visit her in her new world), ...but putting myself in your shoes, I'm not sure either of these two options are really good ones. The hard route to the West is doable, but without significant sailing experience and confidence in your new boat it could become a really bad scene. The longer, easier route would allow you to build your sailing experience and confidence in the boat as you go (and the offshore bits are at the end, not the beginning), but the trip takes too much time. Maybe, the idea of sailing to Honduras just doesn't work???????
An alternative.... Have you thought of taking the money you've saved and buying a really sturdy car / 4WD vehicle and driving to Honduras? That, too, could be a real adventure. Between now and your departure learn some Spanish, take a course in auto repair, learn something about the history and culture of the places you'll pass through. When you arrive in Honduras you'll have the means to let your daughter show you around the country. If she's living on a tight budget, you may provide her with her best chance to see the country. You'll have more time to spend there with her. When you get home you have a vehicle you can sell more easily than a boat if you don't need it any more. In short, it's probably less risky in terms of personal safety (although, driving in Mexico and C.Am is not without risk), and the skills you don't have now that can add to your personal safety and enjoyment of the trip can more easily be acquired between now and the time you leave. You can always buy a boat and go sailing when you get home.
Have a look at Diary of a Journey down the Pan-American Highway
or Google the Pan American highway.
Regardless of the mode of transport, do go to Honduras. You and your daughter will cherish the memories of the trip for a lifetime.
Oh, and I almost forgot....my wife want's to know the breed of your dog.