Is there a wp-config.php option or constant that I can set to allow me to download nightly / beta builds?

There’s not a constant or anything there, but you can install the WP Beta Tester plugin to switch your update stream to the latest nightlies. Once you install this plugin, you can automatically update to the latest “bleeding edge” release from the standard update panel.

Just remember, it’s not advised to update production sites to the latest nightly build. Something might be broken, so always back up before you update.