How do I stabilize a knob on a hollow-core bifold door?
Polyurethane Foam.  The type that is used for insulation.  


Remove the door, remove knob and screw
Tape rear hole to contain foam.
Lay door flat
Enlarge knob hole to allow dispensing wand to fit.  (If this enlargment is too close to the knob base size, this method won't work)
Dispense foam into hole.  Probably a 3-4 second burst will be enough
After curing overnight, redrill the hole to allow screw to pass.


This works because the foam will keep the hollow walls from collapsing