Connecting Google Calendar with self-hosted Microsoft Bookings/Calendly for joint demo slots using callin.io

I'm facing a challenge with scheduling demos for three parties simultaneously using a Microsoft Bookings calendar. Coordinating times via asynchronous email communication, especially when each party is from a different organization, has proven difficult, and I'm looking for an automation solution.

My company resells another company's software, which uses a Microsoft Bookings calendar for demos. I need to book appointments for prospects without sharing the software company's calendar directly, as it's intended for the reseller, not the prospect, to manage.

Is there a method to host my own Microsoft Bookings or Calendly schedule that integrates with my Google Calendar? This would allow prospects to select a time from my calendar, which would then reflect availability from the Microsoft Bookings calendar. The goal is for prospects to see times when both myself and the software company are available for a demo. When a prospect books a slot on my Calendly or Microsoft Bookings, it should also book a corresponding time on the software company's Microsoft Bookings calendar.

I've attempted to coordinate solely through email, but by the time my prospect and I agreed on a suitable time, that slot was already taken on the software company's calendar.

Any advice would be greatly appreciated! I might be able to get access to view or edit the software company's calendar.