Sorry to hear that integration hasn’t gone too smoothly for you.
It looks pretty good (to me) right now and it seems to be the Default Events Template that you are using currently. Is customizing that template to add a sidebar a possibility?
Our themer’s guide goes into more depth, but the basic idea is to copy templates from the plugin directory to your theme. In this for example you might copy:
plugins/the-events-calendar/views/default-template.php
Then place your copy within a tribe-events subdirectory inside your theme:
themes/your-theme/tribe-events/default-template.php
Adding the sidebar is potentially as simple as placing an extra line of code (possibly get_sidebar(), depending on how your theme is structured). Is that something you could give a try here?