Is Calendly the best option for customer-based scheduling?

In my civilian job we are trying to re-vamp access to our faculty. The end-state would be a singular point where personnel/customers can request meetings based on the availability of everyone in the department. Anyone know if there is an option for this?

We need to sync everyone in the department's calendar into one primary group, then create sub-groups. Someone will click an option for a specific meeting request, then availability for the qualifying personnel will pop up.

For example, someone needs to meet about research proposals, so after clicking that option, the three people with approving authority for research proposals pop up on a calendar with their availability.