What I did with the thin stuff is I taped it gently to some aquarelle, folder it over the kozo, prepared my Pt/Pd puddle on the folded part, spread it out and got the rod wet and then pulled the wet rod over the kozo. Seemed to work, more or less. Alas I only had the small amount I got with the the B&S sampler, but I've ordered more. It really has a very nice character. The heavier stuff is fine too and doesn't have the lines.
I've not tried prehumidifying the kozo, I figured it was too thin and loose-fibered. Anyway I think backing paper may be the key. Maybe you could prewet the backing paper and let a bit of moisture seep through the thin kozo?
It's well worth the bother I think, that paper is pretty exquisite.
I wonder about airbrushing the solution!