The most common gateways from the US are Cancun, Nassau and Toronto (believe it or not).
MoBay or Puerto Plata (POP) probably won't be particularly cheap for several reasons. First of all, the airfares to get to either of them will probably cost you quite a bit. I've been looking into flights to Sosua lately and have been surprised and dismayed that it costs even more to fly there than it does to go to CR even though it is so much closer to where I am in Florida. Secondly, the direct flights to Havana from those countries fly from their main airports near their capitals (Kingston and Santo Domingo) not the regional international airports you'd be using thereby adding to fares and flight times. Thirdly, the DR is much further from Havana (~850mi) than either Cancun (317mi) or Nassau (345mi). Jamaica is a little closer (~500mi) but still not nearly as close and it is also much further from the US. Taken altogether that means higher fares and longer total flight times going through either of those countries. BTW, the Caymans are only 270mi from Havana but also on the "wrong side" of the island and a VERY pricey island to visit.
Cancun isn't Mexico's principal airport but as a prime tourist destination close to much of the US has lots of cheap fares to get to and also direct flights to Havana since it is located on the coast closest to Cuba. This is why it is the most popular gateway city for most of the US, including the south central US (TX - LA) as well as all of the western states. Flying to Toronto as a gateway city may seem counterintuitive since it is NORTH of the US and Cuba is south, but it might make sense for those in the northern states particularly the Northeast and places like Michigan, where you fly out. That really depends on whether promotion fares are running from Toronto when you plan to go. For guys in my part of the country, the Southeast but particularly Florida, the Bahamas are usually the best deal since the flight from MIA to Nassau is dirt cheap and very short.
However, as you pointed out, cost is not the only factor for you since you've already ruled out Cancun (and Panama) over much more expensive Sosua, DR. Nassau might not be any cheaper (or shorter flight) or might even wind up costing you a little more than Cancun and offer even less attraction to you than Cancun (similarly, I'm assuming Belize is out). When all gets said and done, SJO will probably be your best bet. It will be much longer total flight times than the other options, but shouldn't cost you that much more than Cancun or Nassau and because it is obviously already one of your preferred destinations will offer you a lot more entertainment value of its own.
BTW, there are ways around the credit card paper trail just as there are ways around the passport paper trail. Some useful/informative websites you might want to check out:
http://www.cuba-junky.com/index.html
http://www.havana-rentals.com/index.htm
http://www.cubatravelusa.com/