As far as I'm aware, campers are allowed everywhere on the Cape, and yours looks like it would handle anything up there, including the Old Telegraph Track and Frenchmans Track... even the Old Coach Road if you're experienced 4wders.
If you're worried about bush pinstriping, the odd dent or turning the camper into a fishbowl at Nolans Brook, you could always base yourself at Bramwell Station or Eliot Falls for a few days to explore the Old Telegraph Track without the camper.
Plenty of people take their campers up there, and the Outback Tanami looks to be capable, but it probably comes down to personal choice.
Why buy an off-road camper trailer if not to take it off-road?
Cape York is a wonderful place. Whatever you decide, you'll have a great time. Happy Travels

Cheers
Karen