I flew the 310 to Belize in 2012 right after the engines were done. In that case I went via Key West. A few notes below:
1) If you're going to stop in Mexico, even for fuel, make sure that your insurance is properly sorted out for the trip. Mexico required special insurance for a while. I think those rules have been relaxed, but talk to your insurance company.
2) Flight into Belize is easy. There one airport in Belize City is the only IFR airport, and all arrivals/departures from the country must be from that airport. You need to use a handler. I can give you the contact info of the man I used. He was great - we got in and out in no time, and the fees were reasonable. You get to cut in front of all the tourists in line at customs, works great.
3) Once in Belize, all other flying is VFR. You will need to get "approval" to go land at other airports, but that was easy enough to get. The handler can help you with this.
4) Make sure to know the size of the runway for the small airports. If you're going in a low wing, make sure to know how tall the brush is. In the case of Placencia (where I went), the bushes were tall enough that the 310's tip tanks were bouncing over them. An Aztec could not have flown into that runway because of the bush height.
If I were doing the trip personally and it didn't add too many miles (and if you don't mind over water in a piston single), then I would go via Key West and Cozumel if necessary for a fuel stop. Really it's a pretty easy trip that way and the over water flying is nice. You have ATC contact the whole way. Bring a raft and a 406 PLB.
You probably will want some local knowledge regarding parking at the smaller airports. I know when I went my handler was trying to discourage me from parking overnight in Placencia. We had no issues, but the people I was meeting knew the people who ran the airport operations down there, and they had guards for the Caravans there overnight. So I think that helped us. I don't have any real gauge there, but it's worth looking into.
Flying in Belize was really fun. I would definitely do it again.