Hey Aaron,
Thanks for reaching out.
The name of the widget’s class to use there is Tribe__Events__Pro__Mini_Calendar_Widget.
However, I would not recommend using the_widget() in this case.
I would recommend using PHP like this instead:
<?php
echo do_shortcode( '[tribe_mini_calendar]' ); ?>
The reason why I recommend this is because if you use this method, you still get to place the mini calendar widget output in a PHP page. However, you also can more easily use the shortcode options (which are covered in detail in this knowledgebase article).
So, say you wanted to filter the calendar by category. You could do this:
<?php
echo do_shortcode( '[tribe_mini_calendar categories="concert"]' ); ?>
☝️ Whereas doing this by using the_widget() is a bit more complicated.
Just some thoughts; if you want to use the_widget(), then again the class name to use is Tribe__Events__Pro__Mini_Calendar_Widget.
Best of luck with your customizations!
Sincerely,
George