I have the 300 f4 and a 1.4x, and unless the bird is large and close, that combo isn't enough.
The extender adds a focusing delay, which doesn't work out very well with moving subjects, plus f 4 now becomes f 5.6. If I understand it correctly, different camera AF points are used for different lenses and f stops.
I asked pretty much the same question you have, and the consensus was to get the lens focal length that works without an extender, and this is 400 to 500 mm range.