I really don't claim to be an expert on anything. I have just had a long fascination with sun/pith helmets. That said, I can comment on a few things.
First, I've never seen anything like this helmet before, but there is a first time for everything.
Tropical helmets were produced by small shops throughout Europe as much for commercial sales as for the military. What is fascinating is the lack of standards and the sheer number of patterns that existed in the 19th century.
As for this particular helmet, it appears to be crudely made, and it has been painted at some point. That doesn’t make much sense to me, but without seeing it in person, I can’t really say why it may have been painted. Blanco and pipe clay were used at the time, not paint.
I agree with John that the one stamp may have been done after the rip was already there. I have never seen ink bleed through to cork so precisely. In fact, I believe the stamps are all fakes, done much later. These are too "official" for a pattern we haven't seen.
If this helmet were $200 or even $300, I would be tempted to buy it as a curiosity and for study. I don't know where it is for sale, so I don't know the price, but I imagine it is much more. It may be a variation of the Bortfeldt pattern helmet, or it could be an attempt to counterfeit a Bortfeldt helmet.
One thought that crossed my mind was whether this was for a past movie production. I've seen replica sun helmets made of cork from the 1960s and 1970s. Even in the 1980s, British sun helmets were made of cork, but today they are made of fiberglass and foam. In my opinion, this is an old fake.