How to install hinges after cutting custom door past wooden edges and composite exposed?
The problem is that whatever composite material you're referring to probably doesn't hold screw threads well or is simply too soft to sustain the screw shank. That's the primary reason for the solid wood insert rail (along with aesthetics and possible surface durability).

You have two primary options:


Use long screws. 2-1/2" or 3" screws of adequate diameter (#12 or 14) should hold well enough. Mortise your hinges in accurately so that the screws don't carry all the shear load. Pilot the screws appropriately to avoid splitting.
Rabbet out some of the composite material and install a replacement wooden rail. You could do this on a table saw with the door on edge if you have a steady-handed friend available. Bond it in place with urethane or high quality wood glue, clamp for a sufficient period, and do your mortises. Pre-drill with the proper size for your screws to prevent splitting.