FWIW I vote for the 400 f5.6. I have experience of both. For many years my bird lense was the older non-IS 300mm. I was very happy with this lense. IS had not been invented, so I didn't miss it. When I saw a 400mm for a good price, I bought it and hardly ever touched the 300mm again, and subsequently sold it.
Shortly after this I came across a 300 2.8 IS in a mega-deal, bought it, and started to use that all the time. A brilliant lense IQ-wise, but, I now realise, basically too heavy. I then sold the 400mm to raise funds, but missed the portability. If out birding, I wanted a camera, but not 4kgs worth. I decided I DID want IS post the 300 2.8, so bought the 300 f4 IS. My pictures were not as good, even WITHOUT the converter, as they had been with the 400mm. I didn't have a duff copy, as sometimes I would get a great image, but the consistency was not there. I sold it again, within about 6 months I think, and bought another 400mm. I use this lense more than I use the 300mm f2.8, which I think is on the limits of portability. A fine lense to be sure, but if I'm birding, it equates to too much kit. I'm a birder first and foremost, it is likely I will sell the 300mm and keep just the 400mm, which as I see it is a perfect birding lense, and no IS is no hindrance.