I really hate to be bearer of bad news (or troubleshoot notes) but the matching issue is quite prevalent on far more mainstream mediums, it seems.
I'm actually using a 6D for an interview while the audio's coming from a Zoom H6.
While I'd like to …
My last two cents would be taking into consideration what Photography literally means:
"To paint with light."
In principle, this is true to videography as well. If you really think with that mind set, then you will see results. Period.
You might want to consider the fact that the GH4 is a very poor performer in low-light circumstances. After the 800 ISO mark, you're going to see noise. The most sought-after feature is its 4K and flat log.
The 4K is then usually downscaled to 108…