I'm not sure how easy it would be to manually adjust to focuser so that the gears mesh perfectly. I suppose it could be done, but it sounds finicky. Not sure about your lever in the middle idea.
I'd try monitoring the linear actuator (or whatever) current during engagement, and the instant it starts to spike (before full engagement), stop and retract, rotate the motor a smidge, and try to engage again.
Also, a bit more complicated, but you could put a spring between the linear actuator (or whatever) and the motor, so that if the gears failed to mesh, the spring would compress (instead of the linear actuator breaking something or stalling). Then when the motor starts, the gears may engage (with a bang)... Also, the spring would have to be stiff enough to sufficiently resist the axial force generated by the motor gear.