I'm documenting this here, just because it's something I imagine I might have to do again someday... and when I do, I want to save myself hours of pain and misdirection.
A client had an old SOAP web service that used IP address whitelisting to authenticate/allow requests. The new PHP infrastructure was built using Docker containers and auto-scaling AWS instances. Because of this, we had a problem: a request could come from one of millions of different IP addresses, since the auto-scaling instances use a pool of millions of AWS IP addresses in a wide array of IP ranges.
Because the client couldn't change their API provider (at least not in any reasonable time-frame), and we didn't want to throw away the ability to auto-scale, and also didn't want to try to build some sort of 'Elastic IP reservation system' so we could draw from a pool of known/reserved IP addresses, we had to find a way to get all our backend API SOAP requests to come from one IP address.
The solution? Reverse-proxy all requests to the backend SOAP API.