I have a pico projector, which I use to display HDMI from my Galaxy S3 via an MHL cable. I recently learned that my phone (and most newer Android devices) can be controlled with USB gamepads, mouses, keyboards, etc. with a USB OTG connector, and I have MegaN64 (a very good Nintendo 64 emulator) on my phone, which works great with my USB gamepad.
I'm trying to combine these two features: I want to be able to provide HDMI input to my projector from my phone, and have a USB gamepad controlling it at the same time. Is there any way to do this? I can't with my current equipment, because MHL takes up the micro USB port, so I can't have anything plugged in at the same time. Even if I had a simple micro USB splitter, as far as I know the micro USB port will not work as a USB port and an MHL port at the same time (PLEASE correct me if I'm wrong).
I know there are wireless HDMI receivers available, like Chromecast, but all of them that I've seen use Wi-Fi instead of Bluetooth, and most only work with certain apps, instead of just mirroring the screen. I plan on using this setup while camping, so sending HDMI over Wi-Fi won't be an option for me.
The only close-to-a-solution I've come up with so far is to connect a controller over Bluetooth: there are a good few apps that will connect a phone to a Wii Remote, but that's not the best controller for some of the games I want to play. I've looked into connecting my preferred (USB) controller over Bluetooth, but I haven't been able find a way to do that.
Is there any alternative to MHL for outputting HDMI which doesn't require Wi-Fi, or a way to connect a USB device over Bluetooth?