All you needed to do was to send a simple email to the property back when the duplicate charge first appeared, asking that the duplicate charge be credited within a specific number of days (I would say five business days). Then, on the sixth business day, initiate a chargeback dispute as you have now done.

It is a complete waste of time and effort to deal with anyone other than the property itself as this is a local accounting issue, meaning that all that is happening is that your emails are being forwarded to the property and secondly, it is a waste of time to make more than one request.

Unlike a service failure where one might question whether good value was received, this one is an easy yes or no.