To the best of my knowledge, BSA (Birmingham Small Arms Company) did not produce a motorcycle that could be converted from a four-cycle engine to a two-cycle engine on the fly. BSA was a renowned British motorcycle manufacturer, active from 1861 to 1973, and they produced a wide range of motorcycles with various engine configurations over the years. However, there is no historical record or widely known model from BSA that featured a convertible engine system allowing for switching between four-cycle and two-cycle operation.
It's important to note that four-cycle and two-cycle engines have fundamental design differences and require specific components and systems to operate efficiently. Converting an engine between these two types would involve significant changes to the engine's internal structure, fueling, lubrication, and exhaust systems. Such a conversion would likely require substantial modifications and would not be practical or feasible to perform on the fly.
If you have any specific model or historical reference in mind regarding a BSA motorcycle that could be converted from a four-cycle to a two-cycle engine, please provide additional details so that I can look into it further.