I would use single sensor mode using 51 AF pts (small sensors) and AFC focus priority.
Single sensor because you are digging birds out between branches and leaves. You don't want to give the camera a chance to grab something else.
Focus priority because the willingness of the camera to fire will give you good feedback on the quality of the focus you achieved. And it should give the camera some time to find optimum focus.
In principle you could use AF-S mode but I like to be in AF-C mode so I can rip off a series of images, hoping one is better than the others. In AF-S mode it may be more like all or none unless the bird spends an unusual amount of time perched.
I do this with a D300, 500/4 AFS and TC14 or 17. I also use an SB-800 on a Wimberly flash bracket on a Wimberly gimbal, and a Better Beamer. With flash there are some interesting exposure options but the light will at least get you a usable image when it is otherwise too dark.
I actually find that with flash a manual exposure of f/8 ISO 400 1/250s usually works unless the bird is in direct sun. In bright light I will raise the shutter speed. That puts the flash in FP mode (set the camera for that!) but it won't matter because the bird is in bright light. If the bird is only partly in bright light then I usually write that off as an image not meant to be taken .
The above is very imperfect but I've never found a fully auto-exposure mode that worked better in the real world. It takes some practice though. You may find shutter priority better, at 1/250s with flash, but then you end up at some absurd stopped down aperture for the best shots, where it would be better to have more shutter speed.
Shooting 1/250s at these focal lengths is not easy but we are bound by the camera sync speed in cases where we need serious distance. You might try 1/320s, where you get only about a half stop loss of light in exchange for a little more shutter speed.
I often spot meter, hoping the bird will fill enough of the spot to get a proper exposure. Arguably Matrix would be better because then the flash can operate in iTTL-BL, which would be better in certain brighter lighting, but not in dark light. Just to say I'm not suggesting any sure fire way to meter. It's one reason I use the manual exposure method I described.
I find that, without flash, the 1/FL rule, even on a tripod, seems to be about right (there being no right simple answer for the shutter speed required). That would put you at 1/400s with a TC14 and 1/500s with a TC17 (you did not specify which TC you use).
I birded with a 300/2.8 for many years but did not spend a lot of time trying to shoot small birds in trees until I got the 500. But the principle is the same, I think.
I would not do this without flash- I think it is that important. Flash is not needed to shoot waders out in the open on water, or certainly BIF, but in the bushes and trees I couldn't be very productive without it.