"Is there anyway that I can connect the new wire to the old wire under the house and then pull it up?"

The chance of this working is best approximated as zero.

If you cut the drywall carefully (like with a rotozip) you can use the pieces removed as the patches to fill the holes and make the repair pretty simple.

Yep, and if you've got access to the crawl space, the number of holes you'll probably need to cut is minimal and probably going to be covered up by the stove itself so you don't need to be the greatest drywall guy to do the patch.