1) Sony's own 100-400 works better than any adapted lens, let alone with an additional converter. Especially compared to coming from an APS-C camera, you'll find the 'bare' Sony lens gives you about as much resolving power as the longer Canon lens on the APS-C body. You can skip the adapter and the teleconverter and you'll still have the detail you're used to.
2) Stacking any teleconverter on top of a mount adapter is generally a recipe for failure if you want AF to be any good. With an IS lens like your 509mm it'll also murder the battery double fast. If you shoot for ~400 frames, less than 1 hour or so, and the birds you're chasing are of the larger and slower variety (e.g. herons, geese, some of the more placid eagles), it'll be fi e. If you shoot more/for longer, or simply faster or smaller/twitchier birds, you're going to want to either give up the teleconverter, turn off IS, or keep to native lenses. (See #1.)
3) The 7D mk III is due out within the next three or four months. It will undoubtedly have more resolution, a deeper buffer, more effective AF tracking, and a faster burst rate than the 7D2 you've been using, and in most cases the 5D4 too. (Good chance it won't quite match the 5D4 for total pixel count, but it'll still beat it for pixel density, which is what really matters.) Best of all, all your current gear will work with it perfectly.
4) If you really want to switch from Canon, and you're not sold on the Sony's 100-400, the Nikon D500 and their lenses will serve you better. I really love the a9, but if 500mm+ is what you want, Nikon is currently the #1 and Canon is #2; Sonys have gotten much better with adapters, but anything of that kind of length—and again, let alone with a teleconverter on top—and the tracking and battery life go right down the toilet.
What I suggest you do as your first move is think about whether you can wait for the new Canon or not, and if not, just rent an a9 and 100-400 for a weekend and see how you get on with that. That native combination is great and easily produces the results of your existing 7D2 and 500mm, but with even better focus and faster burst rate. If that image quality isn't enough for you, rent the Nikon native combo next. You're talking about some serious gear, so it's worth actually trying all the options out before going through the hassle of switching systems. It's not fun spending the time and money switching only to find out you still don't get the results you want/have a whole new batch of problems to deal with that you weren't expecting. As a user of Canon, Sony, Nikon, Fuji, Phase, and Mamiya myself, I could never advise someone gambles on switching systems without spending a good few days testing that system first.