The beeper in a Classic handbox is a single component driven by a single stage of a 74LS14 and Vcc.
If you open the handbox, the beeper is the cylindrical object just above the handbox cable connector.
Flip the circuit board over and desolder the beeper. Done.
A slightly less difficult method is to cut the trace between pin 6 of U6 (the 74LS14, immediately "above" the beeper) and the beeper's "non +" pin.
You could put a switch across your cut (allowing re-enabling the beeper), or just solder a wire to jump your cut for more permanent restoration.
Attachments: two versions of the Classic LX200's handbox schematic, and an image of the component layout on the circuit board.