Can I use construction adhesive to mount a 4' LED Shop light on a Flexicore ceiling?
I would not use adhesive to mount a LED fixture. The LED fixtures I use have large heat sinks on the top of the fixture to dissipate the heat generated. Gluing the fixture could cause 2 problems:


The glue would not allow free airflow on the top of the fixture causing a higher temperature than the LEDs are rated for and this could severely reduce their life.
How to remove a dead glued to the ceiling fixture. 


I guess there could be A 3rd reason what to do when my closet catches fire from a fixture that was not installed per the listing instructions ( ok this one is a stretch because the drivers usually shut down when they overheat).

I would check into flexcore and see the mounting methods available.