Nomads think he is, and not just because of his Lhost body. But He's a human that works a lot for Aleph. He's only in vAleph, OSSS, and White Company (which he is the owner of). So, he might as well be Aleph. Even his personal symbol and the White company symbol look reminiscent of Aleph.
You just said that CB writes a background and then they don't follow it... regardless: make a proposal for the inclusion of Hawkwood in the roster. It will have the implication that White Company (that he owns) will effectively become Aleph ;). I wouldn't complain to an Aleph mercenary sectorial: helps making money ;) (plus with white company there is a good number of nice pieces as well)
And - which is more important - offers plausible deniability. Black ops is one thing, having a front that isn't your own front is a different kettle of fish. However, this requires said mercenary company to remain outside of the officially recognized structures of Aleph. Otherwise the whole plausible deniability goes to hell in a handbasket. So, I believe the White Company won't become an Aleph sectorial. Same way as StarCo won't become a Nomad one, despite everybody and their grandma knowing StarCo is basically a rebranded Corregidor with few cosmetic touch-ups...
Which makes me speculate that the next operations box will be a new Nomads sectorinal versus a new Aleph Sectorial. Both Sectorials were established to address the integration of EI into the human sphere and limit its influence, but in very different ways.
Both can be true! In other words, they follow it when it suits them. I don't think WC would become a part of Aleph seeing as how he and a Danavas are the only Aleph troops. If they made a new Aleph sectoral, I can see him being in it. I'm also relatively sure we are going to see Aleph vs. Nomads. Specifically OSSS vs. Tunguska.