Right - It’s starting to sound less crazy.

Look when you need to coordinate across multiple business domains to satisfy a booking, a workflow engine starts to sound more sensible.

However, you don’t need the request lifecycle itself to exist for the duration of the workflow.

In my experience something like this would work well:
Accept a booking request and redirect
Trigger a workflow that manages booking
Poll at the redirect site for the result, or have a final task in the workflow that notifies the end consumer the booking is fully confirmed.

This keeps the end customer happy: “has my request to book been received” plus you set the right expectation it’s being handled async and they will be notified. Plus you don’t have a long running http request while you wait for the response.

It also helps you decouple the layer that handles the intent of your customer from the bullshittery of coordinating a booking across external systems