I dunked a Binocular Bag with no problems. At the time I didn't realize there was cardboard inside, so I was fearless. Now I would probably only dunk if it really needed it, and would not leave it in the water very long.
I would be a little concerned about your cardboard being partially detached already, and soaking it might not be a good idea. The bag looks pretty clean and well shaped. I'd probably just vacuum/brush out the inside, and condition it well on the outside (a couple of light coats of CPR, buff with horsehair brush, final coat of Blackrock and buff).