Chosun,
You seem to be seeking fully functioning AF with a teleconverter at apertures smaller than f/8. I think that might be very tough to find.
Lately I've started to feel like AF is overrated. Yes of course it is the only way to get certain kinds of action or BIF shots, and those are some of the most compelling bird images we can get, so I'm not giving up on AF and I will always have at least one camera/lens for that.
However for a static target at a great distance like a "nest site across an inaccessible escarpment," if it is so far that you really need significantly more than your current 1150mm equivalent "reach," I would be inclined to go with a digiscoping approach, or my new preference to digiscoping, which is using a small telescope or just the objective end of a spotting scope without the eyepiece.
In my experience, teleconverters work better at close and medium distances. But no matter what the distance, they really complicate and challenge the optical system and the AF, inevitably degrading image quality to some extent. I try to avoid them entirely if I can. And "stacking" more than 1 TC would be a recipe for poor performance, IMHO.
I've tried traditional digiscoping, but I feel similarly about that approach, with both an eyepiece and a camera lens trying to work in concert. Even if you pay lots of money for a system like Swarovski's TLS-APO lens which was designed specifically to work with the eyepiece, you are usually starting out at a fairly slow aperture of ~f/8, with the aperture getting smaller and diffraction degrading your image more and more as you zoom in.
For my highest magnification bird photography, I currently like the "less is more" approach: Use a high resolution astronomical telescope with very little or nothing between the objective elements and the sensor. Get rid of eyepieces and drastically reduce the number elements in the optical train. No more AF to worry about. You'll get more accuracy focusing manually anyway. Get rid of the mirror slap and the shutter vibration. Mirrorless cameras with an electronic shutter and focus peaking work best. A very good tripod goes without saying, but I also usually turn off the in-body image stabilization (Olympus E-M1). Check out the "
Photography using Astro telescopes" forum if you want to give it a try. That would be my approach for something like an eagle or heron nest at a distance. You still have to contend with the stability of the air (always a factor at distance/high magnification), but the image quality would be less compromised by the optics. As an added bonus, you might save money on such a "reach machine," compared to a prime telephoto lens with AF. And you get a great little telescope to pursue astronomy, if you want.
Just my two cents.
Dave