The purpose of bumblebee is to get you support for both video chipsets on the machine. I suppose (maybe I shouldn't) that the HDMI is only available with the Nvidia chipset... So, once you have Nvidia working, you should get HDMI support...
Try using Windows and check to see if you can ID the chipset that's active when you have HDMI running. I can't be of much help there as I am not a Windows user. I also shy away from complicated hardware setups ... Call me "chicken." But it's a lesson learned from experience.