You should be able to change which GPU uses FaceRig by doing this:

Go to Windows's Graphic Settings, then click on Browse. After that, browse your files and select the FaceRig.exe file. Double click it to add it. After you've added it, click on FaceRig below the browse button (we're in the Graphic Settings window) > Options, and a smaller window should pop up. From that smaller window select the GPU that you want FaceRig to use.

If this doesn't work, go into your main GPU's control center and look for the setting that would make it the default GPU.

Let me know how it goes.

I've used that function already, but I could only specify a GPU that was set to the main monitor. GPU 0 (1060) is connected to Monitor 1, GPU 1 (2070) is connected to Monitor 2. FaceRig.exe enabled GPU 0 in Nvidia control panel, but the program still used GPU 1, which is set to the primary monitor.

On OBS, we can specify which GPUs the program will use within the program. Could you add a similar option like this?