You're right about the aperture of the Tamron at 300/600 to an extent, it will be f11 because a X2 TC reduces the maximum aperture by 2 full f stops(f5.6-f8-f11), it's not simply a case of multiplying the max aperture by 2. A X1.4 reduces it by 1 full f stop. I don't think brighter light will make a difference as my understanding was that the camera will not AF with a max aperture greater than f5.6. A 1D will AF at a max aperture of f8. To be clear, this isn't the same as selecting an aperture of eg f11 on any given lens, as to AF a lens will be used at it's maximum aperture and stopped down to the selected fstop at the moment of capture. Hope this makes sense.
That all said, by that rationale, even at 70mm, with the X2TC on, your lens should have a max ap of f8 and therefore not AF, but you report it does. I have read that some 3rd party TCs will allow AF at slower than f5.6, but yours is Canon.

Either way, I don't think it will fit the Sigma, for the same reason it won't fit your kit lens. I wouldn't let this put you off though. 125mm isn't a bad length for general stuff.
Of course, you could just take the gear you already have. If it all fits in the bag you have, and the bag isn't too big of course (I could fit a 35mm EOS with the 28-105 fitted and a Canon 75-300 in one of
these). Seems a shame to have just got your gear and not go make use of it. It covers a good range and the hol will give you a good chance to use and evaluate it.