As a workarround we also tried to connect the Wireless Controller directly to the Hosting Computer, but unfortunately it looks like the NVIDIA Gamestream Server generally prevents any Controller input on the Hosting Machine while streaming.
Luckily, there is a fix for that:
You'll need to rename the DLL that Nvidia is using to send controller input so it won't be used anymore. On 32-bit and 64-bit Windows, rename rxinput.dll to rxinput.dll.old on in C:Program FilesNVIDIA CorporationNvStreamSrv. On 64-bit versions, there's another copy of the DLL in C:Program Files (x86)NVIDIA CorporationNvStreamSrv which you'll want to rename.
After renaming the files, the controller is working perfectly, without any noticable input lag. 🙂