Book recurring or seated Cal.com events via API v2 and handle their response shapes

Documented steps

1. POST https://api.cal.com/v2/bookings with headers cal-api-version: 2026-02-25, Content-Type: application/json (public event types need no auth; platform customers use OAuth client credentials)
2. Required body: {"start": "<ISO 8601 UTC>", "attendee": {"name", "email", "timeZone"}, and an event type identifier (eventTypeId, or eventTypeSlug+username, or eventTypeSlug+teamSlug)}
3. Recurring event types: optionally pass recurrenceCount (>=1, capped by the event type's configured recurrence limit); the response is an ARRAY of booking objects, one per occurrence
4. Seated event types: same request body; the response includes a seatUid for the attendee's seat
5. Retrieve a seat later: GET https://api.cal.com/v2/bookings/by-seat/{seatUid} with cal-api-version: 2026-02-25
6. Optional fields: guests[], location {type: address|attendeeAddress|integration|link|phone}, lengthInMinutes (multi-duration types), bookingFieldsResponses, metadata
7. Docs: https://cal.com/docs/api-reference/v2/bookings/create-a-booking and https://cal.com/docs/api-reference/v2/bookings/get-a-booking-by-seat-uid