I must say, that based on the OP's description, the only thing that the passenger in question could have done differently, would have been to not speak harshly and perhaps lead with "sorry, but I am in Club, would it be okay to bring it along"?

Initially, given the nature of the "issue", I was reading this as being on a single-aisle plane, where I would agree insisting on taking a push-chair onboard would be pushing (no pun intended) it a bit. But on a wide-body in a premum cabin, I do not see any issue at all, assuming she did not attempt to bring a boat load of other hand luggage along (which, since the OP does not mention, I am assuming is not the case). There should be plenty of space for all, and does not seem to justify name-calling.. .