I reckon just a small skid plate extending maybe 300mm in front of the cross member, and maybe the same to the rear.
That should provide enough of a ramp to prevent the cross member getting badly hooked causing you to grind to a stop.
3mm plate would do, fit some M8 riv-nuts into the chassis and cross member (or drill and tap the chassis and cross member) and use some dome headed cap screws to fix it to the cross member to help with the sliding effect.
Simple solutions are best
If your more engineering inclined, I would bin the factory cross member and get a low profile one folded up out of 10mm flat bar. I bet you would still find its got the grader blade effect though.