@t.a.danielsen @equibodyapp is correct that I have a template that can do this.
I’ve built the most complete clone of Calendly for booking meetings with users which uses availability set by the user and cross checks booked meetings to create a list of available time slots on the day they want to book a meeting.
Because of the methods utilized to generate the available time slots is quite flexible, it is possible to customize this to also use Google Calendar events in consideration of availability, as well as customize it to check against multiple users, however, it is not currently set up to do so.
I have also just released another great scheduling template that is built with the sole purpose of booking meetings with various users (multiple schedules) by checking their connected google calendars. Once the desired dates and times are chosen the system checks all selected uses Google Calendars to create a complete list of available time slots across the dates chosen.
It has just been released and you can find the forum post HERE and the template is on the marketplace.
For your use case I believe the newest template would be most suitable as the R&D team, Business Development and Meeting room could all have Google Calendars associated and it will make for a very lightweight but powerful meeting scheduler.
The template was built with integration into existing projects in mind, so the integration process is not too difficult. There are some custom date range picker and time picker that could be swapped out to cut down on integration time.