Just yesterday I launched the launch page for my upcoming calendar as a backend service:
https://blobcal.com/ ! CalDAV support is one of the features I plan to implement if there is demand.
But looking at your current app website, it looks to me as if you don't necessarily require to manage your user calendars. Just getting your app information in their calendars is probably enough, whatever 3rd party provider they are using? In that case https://www.cronofy.com/ might be a good fit.
