Personally I would mounted the ballscrews on the outside of the bed side-frames and drive the base of the gantry legs directly.

This will give a much stiffer gantry drive system, allows you to fill the middle with a nice stiff bed cross members, and allows all of the bed frame to sit on the table (rather than feet at the corners top raise it).

As drawn it's a lot of work and somewhat of a performance compromise 'just to hide the drive mechanism'. If you are worried about swarf on the screws there are brush solutions possible (see a recent machine built by Dean). But on the outside of the gantry sides is reasonably protected anyway.

You can still drive from a single motor by mounting it sticking out of the rear cross member with an arrangement of belts.