It is the classic debate as soon as you start talking supertelephoto lenses. First thing I will say is that there may not be a "right" answer. Just good answers. Very similar to picking "the best portrait prime" you will find lots of arguments for 35 mm, 50 mm, 85 mm, 105 mm, and 135 mm with the occasional person coming out saying the 200 f/2 is the best portrait lens and look at these amazing photos that prove it.... But it gets down to what you want at that moment.
Second, my general impression is that both the 500 and 600 are so amazing, you will likely be happy with either. But Canon makes them both for a reason, they are different lenses. I am happy with my choice of the 500 as it fits my needs a bit better than the 600. For example, this summer I did not set up on backyard birds once. Their, the 600 would be better than the 500. But I did go out kayaking with the 500 twice. I would not want the 600 mm in a kayak. But I can handhold the 500 and my kayak is stable. I did travel with the 500. This is a bit of a push as you can, of course, fly with the 600. But I like the fact that I can fit my 500 and my other gear into a single backpack. Similar to what Joel just described, I have heard several stories of those flying with the 600 taking their gear in multiple bags.
Then there is what else is in your kit. As you say, you already have the 100-400 II, and 400 f/5.6. The 600 is "more different" to those lenses than the 500. As the EF telephoto lenses have come crashing down, I've eyeballed a couple (400 f/2.8, 200-400, even the 600). Had I originally bought the 600, I would have been more tempted by the 200-400 or the 400 f/2.8. But the 500 is essentially a jack of all trades. I actually considered this as I was buying it, thinking I likely wanted a single supertele.
So, it is a big decision. From what you are describing, you already rented the 600 and fell in love with it. Based on that, it sounds like you will mostly be using this for birds, and the fact you have two other 400 mm lenses, it seems that the 600 may be a better fit. But, of course, up to you.
Good luck!