Here's my response to my own OP question: Because I am better equipped for woodworking than for metalworking, my step-support assembly is made of wood rather than metal. The assembly is five pieces of 1 x 4 fastened together with 2-inch drywall screws. The first photo shows the assembly before installation, sitting on top of the scissors-step framework. Notice the shelf that is fastened on top of the two horizontal 1 x 4 cross pieces. When the step-support assembly is in place and the steps are stowed, part of the shelf projects toward you, beyond the front 1 x 4 crosspiece. This projection slides between two of the steps and supports the step above it. By supporting this step, the step-support assembly supports the entire scissor-step assembly. And the steps no longer bounce against and damage the camper as we drive down the road.
The other photos show various perspectives after installation.

The picture below shows the step-support assembly in place. It hangs from the two bolts that secure the original step assembly to the camper. When the steps are stowed, they secure the step-support assembly against the camper.
What I like about this arrangement is that nothing needs to be fastened, unfastened, or picked up and carried when the steps are deployed or stowed.



