You may have already resolved your concern.. but as a contractor, and flooring installer too.. I'd not screw it down. Actually it moves VERY LITTLE with moisture expansion being the biggest factor. But, double sided foam adhesive tape is quite strong for the little amount of retention force that you need. It will not leave holes, can be removed, will not cause your flooring problems, and it will do the job. I'd say it was not only good advise, but the best solution.
If you decide to use it.. clean all pieces first. Attach it to the bottom of the slide.. and align it carefully. When it sticks, it sticks really well!! But push it down.. and give it a day.. it will actually form a stronger bond.
HandyguyJ