Here's a suggestion & its a cheap one.
You've got 2 5D3s & a collection of Canon L glass, yes?
Install a stable version of Magic Lantern for the 5D3 - NO, you're not going to use RAW!
In the Movie menu, at the top, there's this setting that will increase the Bit Rate of the H264 video.
Now because the 5D3 will accept fast CF cards, you should be able to increase the recorded Bit Rate & record sound (you'll have to experiment how far you can push the Bit Rate, but anything over 2 will be good - 3 would be great!).
If you use a good PS (VisionColor is v.good), this increase in Bit Rate will give you a lot more room to CC & even sharpen your footage in post.
Quite frankly, that's the only/best solution for you - a Nex isn't going to be better than a 5D3, not today, tomorrow, never.
Furthermore, for any detailed GVs that you might need, which don't need sound, you could use a short burst of RAW.
Other than that, I just don't see what your dilemma is - you've got 2 5D3s, use them Goddamit!