With a draft of no more than 7' you should have no real problems getting to a berth at BYC, and certainly no problems once there. You'll need expert guidance to come through the reef and get to BYC though, preferably with a local pilot on board. Radios aren't used much here, but there should probably be someone listening on channel 16. If you establish contact you will need them to phone BYC (226-2777) to announce your arrival. BYC may have a handheld radio for dock staff to use (but don't rely on it), but they certainly don't have a boat to meet you or a pilot you can use.
I used to run a diveshop on the dock of BYC and had a high powered radio and often helped incoming vessels, including going out to guide them in. That stopped when BYC staff broke down the antenna and cut off the electricity. They now have no radio service. The dock was sold and the new owners have extensively dredged, but don't expect any of the usual marina facilities. You CAN dock there, but I recommend you try to find somewhere a bit more welcoming!
One other thing. Don't expect or try to fuel up in San Pedro, as the only fuel station has shallow water and you won't be able to get in there on any tide. if you do need fuel you'll have to tender it in drums, which is hard work, dirty and very slow. I don't know about fuel at Caye Chapel. Better to top up in Mexico on the way down - Isla Mujeres is a world-class marina. There used to be a fuel station at BYC which of course was very convenient, but the present owners closed it. They really do believe in looking after their marina guests!
Before you plan on staying at BYC, look up their marina charges on their website. Most people take one look and go elsewhere.
Just spotted you have been given advice by BYC. Beware that the person giving advice may not know the first thing about boats and will say whatever is needed to encourage you to come. Two boats so encouraged have to my certain knowledge run aground and sustained severe damage to props/rudder.