Having worked with musicians and dance people, I know they sometimes have champagne dreams but beer budgets. But, you may be able to do this at a reasonable cost.
You will need a Windows notebook PC, Windows 7 or later, two 3.0 USB ports and two USB capture cards such as this:
http://a.co/hLWhWMv On the laptop, you will need to install vMix, the free version, to accept the two camera inputs and output through the laptop’s HDMI port to the projector. Free vMix will allow two inputs as well as an ability to record to the hard drive. http://www.vmix.com/
I do a lot of live video streaming but have never used a GH4 for that purpose, so you will need to test this out before committing. Essentially, you will be setting up a 2-camera live shoot. These suggestions will make that possible, but there will be an initial investment unless you can borrow or rent the capture cards and laptop.
Here’s a good primer video from vMix: https://youtu.be/FqMEcc3Eo94
