@ondrej
I did read your question from start to finish.
Now what you need to do is try it and see what happens when the card drivers conflict. What you could try is installing the cards using only the 2060 drivers to run both cards. If it fails I would go to the TPU forums and ask the experts there. It's highly likely they will give you the same answer as I did earlier unless one of the members has a magic fix.
What you could try is putting the 1GB card in the x8 slot and the 12GB card in the x16 slot. Then plug in the monitor to the 1GB card. Trying to get it to work is better than not trying at all.