Using the API rather than email is a good way to do this, you’re right, though I’ll check with one of my team members more familiar with the API to say exactly which calls to make.
Or, you could probably modify the headers to be:
Sender: "Web Order Bot" <order-bot@our-domain.com> # (clearly whitelisted)
To: discourse-ingestion@discoursemail.com
From: "New Customer" <never-been-to-discourse@customer.com>
and that should work as you expect (I think, I haven’t actually tested that).