Have you ever tried to modify The Events Calendar plugin, specifically the gcal buttons? The Modern Tribe team does an amazing job documenting their event calendar plugin. I recommend reading over it, but if it doesn’t make sense, then here is a quick snippet to help reposition the calendar buttons before the event’s content. Copy and paste this snippet into your child theme’s function.php file.
Snippet: The Events Calendar ical and gcal buttons
p.s. If you don’t have a child theme, then you should learn how to make one. It allows you to easily add modifications to your existing theme without losing your changes after an update.