TBH, there's not a lot to gain over the 400/2.8 + 1.4x TC (v III). You'll probably need to do AFMA for the combo, and you might have to try a few TC's to find the best match. But you wind up with a 560/4 for a lot less money and very close on IQ in actual use.
The 400/2.8, 600/4, and 800/5.6 can all resolve about the same minimum size detail at the same distance...
I just sold my 400/2.8 AI-S last month. I had it for about 15 years, and like you I always wondered about switching it for the 600/4 AI-S. I primarily used it with a TC-14B, but also used it a lot with the 1.4 stacked with the TC-16A. This is an extremely sharp lens even with the various converters including the TC-300. When I bought it it replaced my Canon FD 400/2.8 L, and it was equally as sharp. I used the Nikon 400 on D2X, D3300, Canon 5DII/5DSR, and Sony A6400. Results just got better and better!
I came across a really cheap Canon FD 800mm f5.6 L a couple years ago and when I got the 5DSR last year I discovered that I could beat the A6000/400/1.4x/1.6x combo with the 5DSR/800/1.4X combo and stopped using the 400. The 400 with TC-300 is just as sharp as the 800/5.6 L, but has slightly more flare.
For the past few years I was finding that I could use the 800mm focal length much more often than 400mm, and was not using it anymore so I chose to sell it.
I suspect from my experience that the 400/2.8 with converters would be as sharp as the 600/4 and 800/5.6. I would stick with the 400 unless you needed to get beyond 1000mm. The nice thing about the 400 is that you can use it as a very sharp 400/560/800/900mm lens making it useful for many situations. When using it with converters I highly recommend stopping it down one stop to maintain sharpness. Have fun with it and good luck.