I don't see how a throttle would care what battery you have, it just needs to hook up to the controller. Maybe if it had little lights to tell you the state of charge, the voltage would be important, but not for the function of the throttle itself.

Sounds like your 36v contoller will work fine. Check your voltage specifications on your charger and your BMS for the battery - you will be doing a lot more than 48 volts. My 36v battery charges at 42 volts. A typical 48v battery will charge at around 55volts.