I believe that I have three choices now:
1) Remove the fretboard, level the neck and reglue the fretboard. Is this a viable thing to do?
2) Remove the frets and level the fretboard itself (at least from 0 to 12th fret, doing the whole lenght would mean sanding off too much, the higher action at higher frets could be bearable I think) – seems like the easiest option to me, but it would involve leveling the binding too and removing/returning the position inlays. I would not like to do this, it feels like cheating and will probably be noticeable.
3) I've been thinking about getting a heat press (a friend could easily build it for me), but I heard that it does not lead to reliable results (even less so that the other options). Perhaps I could straighten it with heat only so that I can install the rod?
The guitar is worth it. Thanks a lot in advance!
The pics:

