OK, so in Build #493 the reason for the crash appeared delayed in stderr:
C:/Users/travis/build/douglaskastle/blender-2.80/blender.exe: error while loading shared libraries: libsndfile-1.dll: cannot open shared object file: No such file or directory
That file (libsndfile-1.dll) is shown in the preceding ls listing, but was not located by the DLL loader. In later builds you correctly cd-ed into ../blender-${BLENDER_VERSION} before running blender.exe from there, which allowed libsndfile-1.dll to be found.
(Although in Build #494 you forgot to prepend the executable file name with ./, so Git Bash searched for it only in the PATH (which doesn’t contain .) and failed.)
Builds #495 through #498 failed due to AVIFIL32.dll not found. This is most likely a Windows Multimedia component unavailable in Windows Server Core. See also this similar issue being faced by others.
Finally, Builds #499, #500, and #502 failed because of the /c argument to cmd.exe – note that Git Bash (just like MSYS / MSYS2) applies its path-conversion logic to change the /c to c: before cmd.exe sees it. To prevent that we need cmd //c.
I’m not familiar with PowerShell, so can’t comment on why Builds #501 and #503 failed silently. My guess at this point would be that it failed due to the same AVIFIL32.dll issue above.