I'm afraid that you may have a tough time finding what you want with your listed requirements. If you are going to use a wood stove then a canvas tent designed for stove use is about the only option I know of.
While canvas is traditionally used with wood stoves, there are drawbacks, being: -canvas is heavy
-canvas tents are expensive.
How about pulling a small toboggan? That might take care of the weight issue. As far as cost goes, there's not much that can be done there unless you find a used tent for a reasonable price.
What about other tents used with modern fuel stoves like mountain climbers etc use? I have zero knowledge of the tents or stoves that they use but someone here will know I'm sure.