I managed to solve this by replacing:
Path.GetDirectoryName((new Uri(typeof(Sdl).Assembly.CodeBase)).LocalPath
with:
Path.GetDirectoryName(System.Reflection.Assembly.GetEntryAssembly().Location);
The former was returning the install location of Mono on the test machine, rather than the executable path where the bundled libraries reside.
Had to do this with the bundled OpenAL libraries as well.