I've tested the 300 F4, 400 f/5.6 and 150-500.
Today I have the 150-500 and the 400 f/5.6.
IMO, it's pretty much like this:
300 f/4 and 400 f/5.6 are both similarly sharp.
If you need 400, the 400 is sharper than the 300 with a 1.4x tele @ 420.
If you don't need 400, the 300 is just as good, and you get IS.
the 150-500 is great for flexibility: it has a big range and great OS. Image quality is a bit lower, but you have to pixel peep. If you don't crop and don't pixel peep, it can be actually difficult to tell photos apart.
150-500 is 500mm only if focused at infinity. If you focus very close, the 400 will give you more magnification than the 500 @ 500mm. This is something not many people talk about.
From 6 meters up, the 500 magnifies more.
I can handhold my 400L at 1/250, but many people need monopods since it lacks IS.
So, I'd resume it to this:
Get the 400 if you need the absolutely best IQ at 400mm and can live without IS.
If you can't live without IS, and can sacrifice a tiny bit of IQ (very tiny), the 300 f/4 plus a 1.4x tele is the way to go.
If you need lots of flexibility and can sacrifice a bit more IQ for that, get the 150-500, save some bucks and be happy. 