I am not familiar enough with that CF7 add on to know if it is possible or not.
In theory… you could create an options page with a field, possibly a repeater, to add dates to. You could then get this list of dates and use this list to disable these dates from being chosen.
You can use the WP functionality for localizing a script to have the list of dates added for your script to use. https://codex.wordpress.org/Function_Reference/wp_localize_script
This would be entirely dependent on whether or not it’s possible in that add on
