Your biggest saving grace at this point is using an AmEx card - they're known for being quite customer-friendly and generally favoring the customer over the merchant during a dispute if the evidence is at all questionable (which is sounds like in this case.) Let the dispute with AmEx play out, and if they ask for any information from you be sure to provide it promptly. If for some reason they don't rule in your favor call back (to AmEx) and see if they'll re-review it.

What hotel is trying to do this? Honestly, it sounds like an owner is trying to squeeze every last dollar out of a property without caring about the customer - sticker removal on a chair usually takes a little bit of Goo-Gone (or similar) and a few minutes of maintenance's time. It's difficult to prove who did it (particularly if housekeeping isn't perfectly diligent) and the reasonable cost of the damage would be, imo, $25 or less - barely worth someone's time to process much less deal with the customer complaints over it, especially if you charge the wrong person over it. If they're charging more than that, I think they're full of it, and hopefully AmEx thinks so as well.