I'll give it a go.
I have both these Lenses, well, I recently sold the 400 as I purchased the 200-400, but the 300f/2.8 II will probably be buried with me.
Large question only you can answer, what will the lens be primarily used for ?? Focal length will answer most of that question for you, do you need to be closer to your subject, can you get close ? Do you need to shoot from further away ?? These sorts of questions will lock in the Focal lLength you will need.
I shoot wildlife, both these Lenses have their place in my shooting style, I've used both the 300 & 400 Version 1 & version 2 lenses, biggest difference is simple, weight, huge difference, the version 2 300 is completely hand hold able, much like the 200f/2, the version 2 400 is hand hold able in short periods, the version 1 was a bit of a brute.
Image quality between the version 1 & 2 lenses in these focal ranges is similar in my experience, but Focus is in my experience much easier to achieve with the version 2 lenses.
Either the 300 or the 400 f/2.8 version 2 lenses will not disappoint on Image Quality, these are simply superb lenses.
If you decide to go 400 you might look at the 200-400f4, almost the same price but with amazing IQ and versatility, but comes at a 1 stop loss of light, which at times I notice, hence my decision to not get rid of the 200f/2 or the 300 f/2.8 II.
Which ever way you decide to go good luck and enjoy.