Like Hank said, there's ton to see and do while based in Hopkins.
There's zip lining just across the highway a little to the south. Not my thing but for some reason unbeknownst to me other people seem to like it.
A little further to the south the Cockscomb Wildlife Sanctuary And Jaguar Preserve is great for nature trails. You can also tube down the river, which your two year old would probably love. As rivers go, it's more like a big creek with a slow current and a shallow bottom. Your chances of bumping into a Jaguar are pretty much less than zero. The chocolate factory is on the way in.
Continuing south, there's Placencia Village, which is a little more developed than Hopkins.
Keep going and there are two Mayan ruins sites. There's also a spice farm where you can ride on the back of a tractor. Your daughter would probably love that, too. If you get down that far, you should stop at Coleman's Cafe for lunch. It's an open air, serve yourself buffet kind of place with great food, friendly service and a very authentic Belizean vibe.
North of Hopkins, go up the Hummingbird Highway and there's The Blue Hole, a shallow swimming spot. The Hummingbird cuts through the Mayan Mountains and is a beautiful drive. If you make it to Belmopan, you can turn right and go see the Zoo and a little further cave tubing and zip lining. Or turn left and go see Actun Tunichil Muknal, not good for two year olds, and maybe go check out hip happening San Ignacio.
There's lots more but those are the highlights.
As far as dive ops based in Hopkins, it's been a while so things may have changed. I think Splash, which is based in Placencia and run by two very good friends of mine, may still have a satellite-type office in Hopkins. If not, you could call them to see if they would pick you up. I imagine there's someone based in Hopkins, but other than Hamanasi's in-house resort operator, I'm not sure. This may or may not extend your bottom time, though, depending on if the group has gone up. I frequently use my own guide because I don't like the herd mentality of constantly moving on to see what's ahead. I prefer to find a nice section of reef and just fin along slowly, watching the fish interact with each other and the reef.
Both Glovers and Southwater offer great diving and are well worth the trip.
Belize regulations require a guide be with you, so solo diving is out. Having said that, you can often arrange to have your own "guide," at least with Splash. What this means is that after you gain some level of trust with your guide, he or she is pretty much content to leave you alone and not ask for your air pressure every five minutes, lol. So it's kind of like a solo dive with a dive buddy along who may or may not point out cool stuff depending on what you want him or her to do. It's almost more of a function of a guide's personality and comfort level with you.
