The short answer is no. The Shuttle works by generating keystrokes that would be automatically directed toward the foreground application. This applies to both Mac and Windows.
On Windows it is possible to write applications that use what we call SDK to integrate get hardware events from the shuttle without having a setting for it. In this case the application needs to process the data and can subscribe to events even when the application is not foreground. Since the SDK is a pain to maintain and support we are pretty quiet about it though.
The shuttle generates keystrokes so, as long as you install the driver on the local client (so NOT inside the remote desktop) there is a good chance that works. At that point it depends on how remote desktop client application treats the generated keystoke and how the timing is.
If it's possible, not sure it is for input devices, you could transfer the USB stream on to the remote machine and in that case work with the driver installed remotely. That is a long shot though and there are way too many "if" I think.