I never post on forums or answer questions, but I just wanted to throw my opinion in on this issue.
I see that a lot of people are urging you towards a BMPCC, which image-wise, is a pretty great camera. One thing to consider with this camera specifically is that the HDMI port is crap. The way the camera is made makes the HDMI port very unreliable and problematic. I shot a feature and went through two seperate BMPCC's with HDMI issues and ended up finishing on a different camera all together. And as another poster mentioned, the monitor on the back of the camera is useless.
Since you will not have access to another camera and you are calling in a lot of favors, I just urge you to think about the reliability of the equipment.
a6300 is a great camera, but has severe rolling shutter, but it is definitely worth consideration. If you get a cage for the camera, it greatly reduces the overheating potential of the camera, as the cage acts as a heatsink. I used to own that camera, and it is very capable of some great images. And you can also get an E-mount adapter to any lens mount out there, so your lenses would work with that camera well.
I just wanted to give you a heads up, because I'd hate to see you get halfway through your film and have a camera on your hands that is almost impossible to shoot with if something does go wrong.