I just had that issue this past summer, and what i did was to get some ABS Plastic Glue from Tap Plastics, and cut up some small chunks of ABS, then glued it to the backside to hold the openframe screen in place. The glue actually melts the plastic, and where the pieces join, turns into one solid part.
Next I cut a bezel to hide the lcd frame. Yours would slightly change the order, where first you would cut a thin 1/32" bezel to hide the lcd frame to fit behind the OEM bezel, then hold the screen in place while gluing blocks of ABS to hold the screen in place.
Go with the Lilliput and take the guts out.
I only needed a few pieces of plastic to hold my frame in place. I used the tape to hold wires and an interface board in place: