Alan,
I think the support bars may already be sagging under their own weight apart from the difficulty in moving them. The leadscrew would be similar plus difficult/dexpensive. I would suggest tubes for the carriage support and looking at tensioned wire or toothed belt for the traverse. Winding the wire around a drum converts length into angular movement and a gear train.
One of the factors in leadscrews is the thread angle. Big lathes have quite course treads eg 4tpi but on a 1.5 to 2 in dia rod. A small lathe might be 8 tpi on a 3/4 in dia. That might give you some idea.