Is there a setting in the plugin backend that automates the booking status update after a specified number of days?

Yes, you can easily automate your customer booking request status utilizing the plugin features. To set up just head towards the General Settings tab and make use of the two toggle options available-

Change Status to Processing automatically updates the status to processing after specified days given in the text box below.

Change Status to Cancelled, which automatically updates the status to canceled after specified days given in the text box below.