How do GLS avionics plot an entire accurate approach path?
The best reference is the ICAO GBAS Guide.  Unfortunately, it doesn't have diagrams to help understand it.  

The Final Approach Segment (FAS) is a geodesic path to the runway. It is horizontally aligned with the runway center line and has a glide path angle that descends to the Threshold Crossing Height (TCH) directly above the Landing Threshold Point (LTP). This can be seen in the following figure:  


Source: SlideShare  

All the necessary data is contained in the type 4 message as the ICAO table below shows:  


Source: ICAO  

There is one Type 4 message for each approach supported by the GBAS ground station.  

The LTP/FTP latitude, longitude, and height are coded in the data block. The FPAP is coded as a delta Lat/Lon from the LTP/FTP. Horizontally, the FAS will pass through both of these points and thus be aligned with the runway center line.  

The vertical component is defined by the Threshold Crossing Point (TCP which is the LTP vertically raised by the TCH) and the Glide Path Angle (GPA).