As I know now, PyTorch comes with all the required CUDA, cuDNN, etc. code bundled in the binaries – I don’t need anything special that would require to compile the sources myself :). Hence, there’s no need to install the CUDA Toolkit at all, and in fact I removed it completely. Only the Nvidia drivers and PyTorch are needed.
I got it working at the moment, simply switching to a  single-card setup, i.e., I removed the “small” NVS 310 – initially, I wanted to keep that card to drive all graphical output and use the 1080 solely for number crunching. However, no combination of drivers or PyTorch version (incl. different CUDA version) worked. With only the 1080 it was smooth sailing and worked immediately. The main difference is, is that I can now a newer Nvidia driver, 415 instead of 390, the limit of the NVS 310.
Your comment that you had to downgrade to Python 3.6 sounds very interesting, though. I actually also tried what happens when using only the NVS 310. I knew that its compute capability was to low, but when I tested it a couple of months ago, I got the respective error messages (“Your card is too old” or something like this). But now, it could even find the Nvidia driver. PyTorch installs quite alright with Anaconda + Python 3.7, it just won’t run in CUDA mode. I will probably give a clean Anaconda + Python 3.6 a shot, just to see if it makes a difference for me. Otherwise, I will leave the NVS 310 out of the machine. I don’t even know, if there’s a serious advantage if the 1080 wouldn’t need to handle the graphics output.
Thanks a lot for your feedback!