Reverse switch for single phase, 2 cap motor

Thanks for letting us know! The sad truth is that even very good quality components can eventually wear out a bit and fail to function exactly right.
A drum switch would still be the better choice, and may still be available from an electrical supply house or a reputable parts distributor.
The big challenge will be to never reverse while the motor is running.
And, if the lathe is like some that I have used, it might be good to add a mechanical link to the brake-bar that would switch the motor off when the brake was applied.
The foot-operated brake bar was both a convenient way of stopping a coasting part and a safety feature, in case the operator got caught somehow. It also moved the control ever to the off position somehow..