Digging into this hand-waved solutions usually led me to nothing. Whenever this or similar problems are mentioned, there's a lot of hand-waving coming to a conclusion 'it just works'. I know this topic is sometimes coming back like a boomerang but having spent literally hours I didn't find anything I can apply, without having any doubts. I'd like to start thinking about HDMI, only if there's really no other good way to do this.
It's such a huge headache, I've started looking into other ways. Technically using HDMI is possible, but it may be extremely hard and expensive to not make too much mess. I'm going to have a setup where it's not easy to connect PC to TV using any available direct display connection.