No matter where you stay, some travel will be involved for all the activities listed. The most central location for all those activities is Belize City, but it's probably the least desirable place to stay in Belize (I've heard it described as a "crime-ridden slum"), and you would still have to do some travelling for each activity because none of your activities are really close to BC.
So, what's the priority? Which activities will you be doing the most?
If diving/snorkelling/beaching is something everyone wants to do most days, then stay at Caye Caulker, Ambergris Caye, Placencia or Hopkins, and do day trips for other activities. If diving/snorkelling/beaching is something you will do only a couple of days, then stay at San Ignacio, Placencia or Hopkins.
Placencia or Hopkins might be your best choice -- great beaches, but the reef is a long boat ride and Mayan sites are a long road trip. Cockscomb Wildlife Sanctuary is close enough for day trips.