Paul, you have pro's & con's on both sides.
A fast focus speed is a must-have for BIF photography. By mounting the 300mm on a 2x extender you will reduce its focus focus speed. Will it focus slower than the Sigma Sport? Not sure, but most likely.
The 2x extender will turn your f/2.8 lens into an f/5.6. The Sigma will give you f/6.3 @ 600mm but that is only a 3rd fstop of light less than the Canon.
Now and again when shooting large raptors in flight (or perched birds) you will find yourself in a situation where you want to zoom out a little (because they are too close). Usually when I'm waiting for a bird to take off I also zoom out a little to provide some space in the direction the bird will most likely fly to. You will not have that option with the 300mm.
The 300mm undoubtedly gives you higher quality images than the Sigma. But will that still be true after adding the extender? Your guess is as good as mine.
So, all considered: I'm leaning towards the Sigma for the purpose you need it. I've used a Nikon 300mm f/4 with a 1.4 TC before I bought my Tamron 150-600 G2 and the only thing I'm missing is the 300mm's faster AF speed. But that should not be an issue with the Sigma Sport which was built with a fast AF speed in mind.