Hopefully this will save time for anyone who ran into the same problem that I did:
After installing the wine-gl extension, I created a custom extension with an additional .dll.so for the /usr/local/lib/wine directory.
However, to my dismay, Wine treated it as if it didn't exist, even though the file was clearly there.
It turns out to be an artifact of the way TC mounts extensions.
Wine seems to default to using a relative path ../lib/wine/ from the executable to search for built-in DLLs.
What this means with TC is that the resultant path is:
/tmp/tcloop/wine-gl/usr/local/bin/../lib/wine/
In other words, it winds up referring to the wine-gl mount rather than /usr/local/lib/wine, and only sees the .dll.so files provided by the wine-gl extension.
The workaround that I used was to "export WINEDLLPATH=/usr/local/lib/wine" before invoking wine to run an application that needed the extra .dll.so file.