My current thinking is to cut the leadscrew shaft (this would reside behind the lower portion of the control box) and bore a 1/4" hole in the cut end of each segment. Ream the holes to end up with a nice slip fit for a piece of 1/4" drill rod. In the short piece of leadscrew shaft, toward the gears, hold the rod in place with a roll pin, extending out from the surface on each side of the leadscrew shaft enough to provide a dog for engagement. On the longer section of leadscrew, toward the carriage, another roll pin through it, but past the depth of the inserted drill rod, which now acts as a pilot shaft to hold the rods aligned as they rotate freely from one another.
Because the length of the leadscrew needs to remain a constant, the material sawn out of the cut will need to be replaced, so a bushing washer will need to be made to fill that gap, turning freely on the pilot shaft.
Next, make a sleeve that is reamed to a slip fit over the surfaces of the adjoining shaft segments. In that sleeve, mill a slot from one end on opposing sides that will fit to span the two roll pins as it slides toward the gears, providing the dog clutch engagement. It will need to be long enough to have abundant material toward the right end of it, unslotted, so it will be strong enough to keep the sleeve slots from trying to widen under the load of engagement. Round the inside corners of the open end of the slot to assist engagement as the sleeve is moved.
I need to do some measuring, but I am hoping to have enough clearance on the unthreaded section of the leadscrew to park the sleeve when disengaged, and have some way to slide it (lever, knob, etc.) protruding out the right end of the control box, yet not get into where I have my carriage stop clamped to keep things from hitting the chuck jaws. I use an aftermarket 4" chuck, which is rather large, so there is a lot of leadscrew below it that never sees use. If necessary, I may have to allow the back end of this sleeve to cover some of the unused leadscrew threads when it is disengaged.
Depending on how these dimensions work out, I may have room to put a yoke lever on the RH end of the sleeve to move it. (Like a clutch yoke on a manual transmisson car.) Otherwise I am thinking a sort of top hat shaped knob on the end of the sleeve, to pull it right or push it left. I wonder if such an arrangement would stay where it is set while using the lathe, or if a detent system would be needed to hold it in each engaged/disengaged position?