I find it amusing that your range warns you the cooking ring will work as intended. It's supposed to be heat trap, and focus heat on the bottom of the wok.
That said, they are also correct that it may discolor the burner grate. I can't really say what your grates are made from, and many cooking materials discolor at high temperatures. I think the main idea behind this paragraph in your manual is to save them from warranty replacements for discolored grates.
I also find it highly suspect that a wok could damage the burner itself. They are usually ceramic, in direct contact with open flame, and any pan traps heat to some degree. Burners get HOT and have to be able to take it. Although the little decorative cap on the burner might discolor as well.
Camping stoves are generally not safe for indoor use. They can produce fumes or carbon monoxide that would be fine outdoors, but dangerous indoors. Check the warning labels before buying anything to use inside.
As for the actual cooking, I don't think that you'll get the results that you expect.
My experience includes an MSR Whisperlite International backpacking stove (white gas), and Coleman two-burner stoves in both white gas and propane variants. Anecdotally, the backpacking stove has limited control, while the two-burner stoves don't quite have the oomph of a real gas stove.
The Coleman links that I provided indicate that the white gas stove has burners that put out 7,500 and 6,500 BTU, while the propane stove puts out 10,000 btu on both burners. Using the REI-provided time for boiling water, I calculated that the backpacking stove puts out about 4,500 BTU (and other backpacking stoves indicate similar times, regardless of fuel).
By comparison, my consumer-grade natural gas-powered kitchen stove has two burners that put out 15,500 BTU, one that puts out 9,500, and one (the simmer burner) that puts out 5,000. Viking offers normal burners up to 18,000 BTU, and a wok burner of 27,000.
However, white gas stoves aren't okay indoors. They're fine while they're burning, but lighting them is a bit dramatic, and once you shut them off they'll put out half-burned fumes for several minutes.
Similarly, propane camping stoves usually put off too much carbon monoxide to be safe indoors.
Best Answer
The larger ring will transmit more energy, but spread over the size of the ring; so the density is probably roughly the same.
So using an element bigger than your pot/wok bottom is basically just a waste of heat/energy.
Some supporting evidence added by DMA57361:
I've recently gotten a free energy monitor, which tells me the current power consumption of my entire home. So, with a bit of experimentation, I have worked out the approximate power output of each ring on my hob, and this backs up that they have approximately the same output density.