Why do people act like the Activision part of Activision Blizzard has any say in what the Blizzard Entertainment part of the company does? Activision is the smallest part of the company and would no longer exist if it weren't for blizzard and King (Blizzard is currently sitting on larger profits than king, which may be the the first time since activision blizzard acquired king? don't quote me on that though).
In short, there is no way the Big Dog is being bullied by the Little Dog, that's just delusional. Also, it's good for the company as a whole to get Destiny 2 out on the launcher, people who may never try Destiny if it were elsewhere may try it, and it may bring Destiny fans to Blizzard games.
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Just for the brand logo, for verbal conversation it'll probably just be referred to as Battle.net. Just like how Blizzard's full Brand logo is Blizzard Entertainment but it's hardly ever referred to in full.