How can I replace the background of my scene, but keep the reflections and lighting generated from my HDRI?
Enable Transparent in Render settings > Film:
Now the sky will render as alpha transparency when seen directly, but still illuminate and reflect off other objects.
Once you have a transparent background, it can easily be filled in with another image via an Alpha Over node:
The Translate node is used for some tweaks to the positioning of the background image.