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I don't have a 2x TC III, but I often use a 400 2.8 IS + 2x TC II on birds, most of the time wide open. AF is slowish, but IQ even wide open is very decent.
From all the comparative tests I've seen, the 2x III is at least optically as good as or a tad better than the 2x II. Thus, I believe that the following examples might have been a bit better or at least as good if I used a 2x III instead of a II.
Lowland White-eye (Zosterops meyeni, a near Philippine endemic, 4 inches total length)
Canon 1D MIV + 400 2.8 IS + Canon 2x TC II, 800 mm, f/5.6, ISO 1600, 1/320 sec, 475B/516 support, manual exposure in available light, near full frame.
Canon 7D + 400 2.8 IS + Canon 2x TC II, 800 mm, f/5.6, ISO 1600, 1/320 sec, 475B/516 support, manual exposure in available light, uncropped full frame.
Canon 5D2 + 400 2.8 L IS + Canon 2x TC II, 800 mm, f/9, 1/250 sec, ISO 200, bean bag, uncropped full frame resized to 1200x800.
Link to 21 MP full res image of the above: