Hi, with this design you could have a fairly heavy flat plate across the X axis and bolt two thick wall pieces of angle iron to it......you don't need to go too high with the angle iron sides, just as wide as the linear rails is sufficient.

The idea of having higher sides to increase rigidity of the cross beam I think is not warranted......if you have a thicker base piece and heavy angles it will be rigid enough.....nothing will act against the material cross sections to make them flex or bow in the middle unless massive forces are envisaged......with a 10mm base plate....not even then.

Having two angle irons back and front for the linear rails, bolted to the plate, will allow you to adjust the width of the cross beam to get it dead parallel and square to the Y axis......which is impossible with a C channel unless you have it machined on three faces and hope it doesn't warp.
Ian.