We do not have a public availability end point. You would need to use Google Calendar API to request this, then pass in the information via our API for the booking date and time to book.
@Ben YCBM has a wonderful API, but I think this is an oversight. I would encourage you to expose the necessary model. I think it makes your API more sticky, because ass you mention yourself, this completely removes the need to use any other API from other calendars. In a sense, if you allow for exporting of availabilities, you are an aggregator for calendars.