Creating a heated chamber from a very large heated bed
Cardboard is fundamentally paper. Paper ignites - just as Ray Bradbury claims - at around 233 °C or 451 °F.
A thin heater wire usually glows red hot under operation. Not dark red, not blood cherry, not dark cherry, it's usually medium cherry to dark orange in operation, as one can easily tell by looking into a trusty toaster oven, which also shows us how powerful of a heating element this is.
In a typical toaster, heater wires are mounted in a metal frame and backed by some non-flammable material that looks like paper but is not. In a 2004 patent for such a method, they use a sheet of mica. But how hot is our wire? Well, under mains power, the wire of a toaster reaches 700-900 °C within seconds, heating the inner chamber of the toaster despite the slots to about 200-250 °C so quick that the bread roasts deliciously... until the heat controlling bimetal strip releases it up.
We operate these wires for at best a few minutes. In an electric forge, pretty much the same wires with a little more thickness to them are run at lemon glow for hours... and they are embedded in firebrick material to prevent them from igniting stuff touching them by accident!
How we know the temperature? Incandescence charts of course: 
Conclusion
Paper or cardboard backing on heater wires is a bad idea. Mounting the heater wire to a metal frame that is as nonconductive as possible and without a chance that something accidentally touches it is needed. Fiberglass or rock material would serve as a good carrier, an aluminium block with radiators would give a good heating-to-surface mix, especially is precautions are taken to make the wire surface nonconductive.
Even using wood close to such a heater is a gamble I would not want to take. It's also best to mount the heating element into an enclosure that can't touch the wooden outer chamber and that stays put if the enclosure is opened or removed.