Some crazy Rube Goldberg thoughts:
0) Convert the brass nut on your slide to a half nut, just like the main leadscrew. One could slice the nut in half and use precision dowel pins to guide the bottom half vertically to maintain alignment. A cam lock would move the lower half up and down.
1) Make a friction clamp for the handwheel. Attach a shaft on the outboard side of the slide with a friction clamp to engage the taper attachment. Friction clamps govern which side has control. Unclamp the handwheel and clamp the taper attachment and you're tapering. Reverse the operation to restore handscrew control. The handwheel clamp allows rotation of the handwheel to turn the leadscrew, and not much else. The taper clamp attaches the taper attachment firmly to the leadscrew, and therefore the slide. The leadscrew stays rigidly fixed to the slide.
2) Build the modern equivalent of a hydraulic tracer lathe. You'd have to be reasonably facile with electronics to do this. I'd attach a step motor to the end of the leadscrew opposite the handwheel. In normal use, the step motor is powered off. You get a slight cogging feel in the handwheel which can be done away with by adding a friction clamp to the motor that is normally loose.
The step motor is driven by a precision limit switch that rides in a track. That track is your taper attachment. The switch rides along, and so long as it is closed (i.e. in contact with the track) nothing happens. If the track moves away from the switch, it opens, and this triggers the step motor to turn until the switch has closed again.
3) Convert the lathe to CNC. Amazing what a lot of attachments are no longer needed with CNC. Taper and ball turning just for starters.
Options #1 and #3 seem the easiest to me.
Cheers,
BW