Add Online Booking to Existing Website - Google Calendar Sync

We have a beauty studio with six specialists and a website we are happy with. React in front, Node and PostgreSQL behind it, and an admin panel we already use for prices and texts. The design stays as it is. Our bookings live in a shared Google Sheet. Clients cannot see what is free, so they call or write and we answer the whole day. And when two of us edit the sheet at the same time, we end up with two clients on the same hour. We want the booking inside our own site, handled in the admin panel we already have. A client picks the service and the specialist, sees only the free hours and books. The same hour must never be sold twice, even if two people press at the same second. Every specialist has her own working hours and days off, and services have different durations. Most clients book from the phone. Each specialist keeps her own personal Google Calendar, separate accounts, we do not have one studio calendar. A new booking first comes into the admin panel, and when we confirm it, the system must put it into the personal calendar of the specialist it was booked for. If we then cancel or move it, her calendar must follow. Until we confirm, the hour is already taken for everybody else - we do not want to confirm and find out it was sold twice. In the admin panel: the day of the whole studio, confirm, cancel or move an appointment, set holidays and block time. Our office manager should run it without calling a developer. No online payment, no SMS, no client accounts. A confirmation and a reminder email if it fits. Budget 1200 USD - if something does not fit in that, say so and write what you would leave for later.