Well,
@RBenton I have done this thing or similar several times, all pretty successfully. While it helps to have the countertop off, it's not always necessary.
What I did in each case is stub off the pipes in the crawl space, in some cases not exactly below the area but a bit of a distance (a couple of feet?) away. At the stub offs, I placed ball valve lever style shutoffs; this is with ½" copper pipes. Remove all the pipes that come through the floor.
Then, I dry-installed the base cabinet, (not affixed to the wall or anything yet) and drilled new holes through the cabinet base, exactly where I wanted them. In the couple of cases where there was tile (porcelain) underneath, after drilling the holes in the cabinet, I took a Sharpie on a stick, and marked through the holes where on the tiles the companion holes need to be; you need to ensure from below you are not hitting a joist. So, be judicious in where you need the pipes to come up.
I then removed the cabinet, drilled the appropriate holes through the tile and subfloor. I then inserted a length of ½" pipe in the appropriate holes, went below, took measurements, cut pipe and sweat soldered it all up. In two cases where I couldn't match 100%, instead of rigid tubing, I used a small length of flexible copper; thankfully the local ACE hardware sells it by the foot, and as I recall I only needed 2'. The big box stores sell it only in 25' coils.
One of the last projects I did like this was a bathroom vanity against a garage wall; the original pipe was in the wall, and had gone up from the crawl space up and over the sill plate on the foundation wall into the finish wall. It was way too cramped to attempt anything in that tight space, so I actually brought the hot and cold lines up the left side (not the back) of the vanity cabinet. It all worked well.
This last job was in my former weekend home in central Michigan. The crawl space there was unlike most crawl spaces I've seen here in the south. Mine was a clean, dry, well lit space with a concrete floor and level. Basically it was a nice basement albeit with a 40" headroom. It's a lot nicer working in that crawl space than what they create around here. Hope yours is as nice.
Good luck!