I have compiled your PAPIs along with the stock PHBK. Both the lights and the 3D models display as expected.
I suspect the issue lies in the scenery priority. Here's my reasoning. Stock airports are loaded first, then addon scenery. Hence , in my test, the AFLT lights are of higher priority and display normally. What's different in your case is that you have two addon packages, the one containing the airport, and mine in the Prepar3d v4 Addons folder. The SDK is not clear about which would have priority. I suspect your airport is of higher priority than the AFLT models (which are in the file I asked you to check) and that your airport contains some exclusions one of which is clearing the 3D PAPI models. The lights would not be affected since they are rendered as SimObjects, not scenery.
I'd like you to try 2 things, please. Move your airport file into your Documents\Prepar3D v4 Addons\AFLT for P3D\Scenery\scenery folder and preface it's file name with "_" to ensure it executes first. If my reasoning is valid, you should see the models. Then, reverse that procedure, i.e., move the file "AFLT_projectID_Bases_X.bgl" into the scenery folder holding your airport. Again, preface the file name with "_". That too should result in the models being displayed.
In the meantime I'll fix a little bug I seem to have introduced into the last release, i.e., PAPIs appear offset both from the runway edge and runway end the same distance as the specification for runway end offset.
Don