I just figured I'd put this here for anyone else who runs into this problem.

Problem description:
* Trails in the Sky FC, SC, or maybe 3rd (haven't tried 3rd) is chosen to be installed via GOG Galaxy.
* Once all files are downloaded, the game will start to be configured.
* A UAC window will pop up requesting elevation for Direct X.
* Upon elevation, an info box pops up with the title "Direct X" and the notably unhelpful message "Invalid command line parameter."
* Galaxy then repeats the last 2 steps (UAC, error) after about 15 seconds.
* After 2 attempts, Galaxy skips to the next game in the download queue, looping back after all other downloads/installations have been completed. The game files are not deleted, but the installation is considered "failed" and the game is not playable.

Problem solution:
While the rest of the download/installation can take place in a standard user session, you seemingly must be logged on to your PC as an administrator to complete the installation. If you got this far on a standard user account, just log in as an admin, open Galaxy, and the installation should complete almost instantly.

My specs:
Win 7 Pro 64-bit SP1
Intel i7-6700HQ @ 2.60 GHz
8 GB memory