I think that you may be headed in the wrong direction here. Those 600 grit discs are described as "for stone" because they are being marketed at those who are polishing stone.
I am not an expert on redoing floors, but I have refinished 4 (or 5, can't recall exactly) with pleasing results. All of your sanding on the floor boards should have already accomplished the surface flattening you want prior to the first coat of varnish.
After the first coat of varnish, you would have found that the grain was sealed and there might have been a slight raise/rise here and there. I've found a light sand with around a 120 grit can be helpful if there are obvious problems, in particular if you found that you failed to really clear all the dust prior to that first coat. Then, give it a good wipe down with a damp cloth and lay on the second coat. (No real harm if your first sand is after the second coat.)
Between the second and third varnish coats, I've never used anything other than around a 240 grit. The goal is much like with gloss painting, which is to flatten, de-nib and provide slight key for the next coat. And, again, like glossing, you then need to wipe down with a damp cloth and follow with a tack cloth if you have some.
Also, I wouldn't personally be looking for sanding discs. Even with fine grits, I find it is just too easy to dig in an edge and wind up marking into the surface when using a disc sander -- they are just too aggressive as they spin. If you want to use power tools, I'd suggest a lower powered random or orbital sheet sander. Don't worry, even by hand, the varnish will come away easily enough with the sand paper.
Anyway, others may do things differently, but this has worked well for me in the past.
Enjoy your floors!