I don't think I'm conig up with anything new here -- the way to do that is to provide countersunk holes in the top of the base; flathead screws (no, not screws with straight screwdriver slots; if that's what comes to mind, look up what flathead really means) would hold the base in place; then a cover held in place by double stick or some other kind of removable but sturdy non-screw attachment would go down over the entire top surface.

Or threaded blind holes in the bottom of the base; it would be up to the installer to drill matching holes through the counter and screw in from below. If either of these ideas is used, that is at least a $250 contribution.