Did you escalate beyond a front line CSR to get this response?
One frontline rep knew this, but several others said the opposite. It was a struggle to have the same person answer consecutive messages. In the end I went into robot mode and auto-replied "I will only respond to messages from manager X", and eventually the messages made their way to her.
By the time we spoke on the phone the points had posted. (They posted backdated to when the reservations had been made, which means I could not see them unless I navigated back a couple months.)
I raised the question during that conversation, and she confirmed that we could pay with "any" card, but that if the points did not post automatically it would be "difficult" for Chase customer service to help.
Bear in mind, I have made several purchases with Citi cards and these have posted automatically. It's only if something goes wrong on the merchant's side that you even need to address this.