I've been to places such as The Cape, across the Simpson and up through the Gulf and never gotten a flat tyre yet, so if it was me i'd just take the 2 you already have on the vehicle and camper, aswell as a repair kit for plugging holes...and know how to use that repair kit!!!
Maybe i've just been lucky on my trips, but the fact i go away with new or near new LT tyres fitted, aswell as deflate the tyres when i hit the rocky sections, i'd like to think plays a part in the no punctures thing. I have gotten a flat tyre on my old Hilux whilst outback many many years ago, but those tyres were barely street legal let alone right for travelling outback roads on IMHO...lol
in saying all that...the extra weight of the carcass is minimal compared to what you'll already be lugging around, it wouldn't cost much to get it fitted onto the rim in a town somewhere should you destroy one along the way. I'd think the chances of you detroying 3 tyres between towns would be pretty remote unless they were crap quality tyres to start with, or you didn't deflate them for sharp rocky roads. If you're not a fan of deflating the tyres for such terrain (some people aren't), then see how it goes on the trip, worst case scenario if you destroy 2 tyres in one area you could always deflate them then and hope to get out of where you are without a spare, or take the ones from the camper to go get help or even drag it slowly on the rims as Speewa has said if you're concerned about leaving it behind.