Attaching closet door divider to laminate
Drill oversize holes (3/8" to 1/2" or so) just through the flooring that are still concealed by the divider's base plate, and mount the divider directly to the subfloor. You may want to use nylon or metal bushings to hold the divider at the same level as the floor (to avoid pinching the flooring), and you may need to purchase longer screws. 

Use a sharp bit, ideally a "pilot point", and a high bit speed with low force. If you don't have a pilot point bit, start with about a 1/8" bit and progress up in size.