Digging out a crawl space is doable. A guy with a frame over brick/block house in my neighborhood had his 3' crawl space dug out for a 9' full basement. A small backhoe dug holes for temporary footers, one at a time, around the foundation. Then he undercut the footer to the inside edge with a shovel. Concrete was poured for these new "big" temporary footers. Once they were done - don't remember how many, but at least one in each corner and maybe one each, midwall- enough to support the house, he punched a hole in the foundation wall and had a small Bobcat dig down until he could get under the sole plate and excavate the crawl space and. He also dug out for new drain tiles and a sump pump well (wet soil area), poured new footers, and installed block up to bottom of (inside?) the old footers, then poured a new basement floor. He did it fairly cheaply- friendly labor and a lot of self-help. It made a mess of his yard and not sure what he did with all the dirt. He also lost about 1-2' of floor space around the perimeter because the new basement walls were located inside the inner edge of the old footer. I think you would need a "certified" engineering plan to get approval to do something like that.

You might also be able to dig out the duck space, by punching holes in the foundation and install temporary I-beams to support, without actually jacking, the house.