This is an interesting customization. I am not sure how much help we can be, but I will see if I can provide some information.
Option 1
I’d like to know if it’s possible to override the widget-calendar.js
You could wp_denqueue_script that script and then replace it with your own in your theme or a custom plugin:
wp_denqueue_script( 'tribe-mini-calendar' );
That looks like the easiest way to achieve this.
Option 2
Another option could be to put a transparent overlay over the calendar widget(not over the nav buttons) that acts as the button for colorbox and then when the calendar changes it happens under that.
Does that make sense?
Thanks for creating the other ticket about the php notice. I am looking into that with the developers.