Another option is to forgo coating the frame altogether. I don't think I have ever coated a kayak frame with anything, except perhaps in the cockpit area, for looks.
And, tung oil is not a particularly long lasting or effective wood preservative - and one that you won't be really able to renew once the skin is on, in any case.

I do oil or varnish the frames of open boats. But, since this is primarily for aesthetic purposes I avoid coating the outer surface of the frame. The outer part is going to be covered by the skin, and I, like you, prefer not to have any bleed through. Saves a little work, too.

Good luck with your project!