What tool is used for this tight plumbing situation?
If you haven't already, the cross-hatched drain cover in the tub can be removed with this wrench, called a bath tub drain wrench (or similar):


There should only be plumbers putty underneath it, so it will be hard to turn at first, but then quickly becomes quite easy.

Once that is out of the way, running a small drain snake down the drain is comparatively easy.  There is probably no need for you to go under the floor.