for this price range, I would go for a canon with magic lantern. I personally have a rebel T2I and it really does the job... more recent ones t#i might slightly be better in low light (check that cause the t2i is only usable until 1000 ISO) but it does a heck of a job and you can find some for a very tiny price. maybe 300$ then you have 300$ for a very good lens, like a 50mm f1.4 which will also make a huge difference (and you won't even have to worry about low light !)
So of course the bit-rate or format aren't ideal, but one cannot compare that with what you get with a 2000$ cam ...

Yeah I would totally recommand the GH4 which has a better bitrate than the sony, and it can really make the difference...