If the switch box is like mine, it only switches RF and Video signals, not HDMI. Mine is a Quest model something...
You could probably make it work if you found an HDMI-to-VHF modulator and converted HDMI to VHF before going to the switch box.
While VHF won't give you the nice picture that HDMI will, consider you are in a motorhome and well... you are "roughing it".
You could probably find an HDMI switcher, but then you would have to run HDMI cables to your other TV locations - which seems like a lot of work.
I think also the switch box will input Video/Audio and has a VHF modulator built in, so if your satellite TV has video and audio jacks, you could probably do it that way too. I plan on installing a DVD player that way as soon as the outside temperature gets above 10deg, but I think that is going to be awhile. So, I am speaking from theory rather than actually doing this myself.