I found some ink that I can use and coming weekend I'm gonna get my some little bottles. It is Noris 110S (or also known as Colop 801) and it is a waterbased ink that is especially for applying on bare wood. Of course I'll do some tests on pieces of scrap wood (not that plywood I used yesterday).
I have installed the tuners on the headstock just to see how it looks and I am perfectly happy with it