There are a couple of things I would consider, and do.
First of all though can you answer one question, regarding the lens, is it exactly as you have described, or is it the macro, HSM or HSM2 version? There are about 6 variants of this lens, and until the latest second version of the optically stabilized lens, the sharpest at maximum focal length was the one you have, or the non DG version of this lens (which came out prior to digital lenses- the only difference is the rear element coating)
Second, I would , especially if it is the lens you describe, add sigmas 2x tele converter. The quality is still more than what you need.
Third, I would gladly trade off shutter speed for f stops, because at 2.8 the DOF is quite shallow, and spot on focusing is a necessity, especially when cropping in to the middle 10% of the frame.
Lastly, and this is a real issue for birders specifically when shooting small birds, get closer. While many (self included) continually state that thee is no substitute for extra focal length, consider this
Image size = subject sic ex focal length / distance
Extra mm of focal length cost a lot of $$, moving closer costs you nothing. I realize, however there are limits to getting closer, because the subject is likely 10 meters up in the tree and you do need some angle on it.
One last point, the shot seems to be darker than I would like, and I wonder r what metering you used
Here is my shot using the earlier APO 70-200 F2.8 EX with the sigma 2x
The lens and TC is the way to go