Solved!
It turns out it WAS a dll issue. It doesn't matter where you put your GDAL dlls, because ArcGIS Explorer uses GDAL, so it is pre-loaded. This is why you are seeing the issue. It seems like it can't find the dll, but it's really a conflict where it can't load it. The solution is to figure out which version of GDAL that the application is using, and use the same version. Unfortunately, it is often difficult to find the .NET wrappers for varying version of GDAL. One direction you could go is to get the source and compile it yourself. I can tell you, this is quite the pain. If you're interested, I can provide you with the dlls that I compiled, which match the dlls in the most recent version of ArcGIS Explorer.