Two years ago, I knew this would happen: no booking website will be happy with an automated agent of any kind which is the user’s agent. They will want it to be their own agent.
And the reason for it is that a good agent will just bulldoze its way through all dark patterns and ways to upsell you. And the airline or hotel (speaking from booking.com experience) want to upsell you so badly it hurts.
Which is why any agentic setup, except for a completely self-hosted one, is not viable in this situation.