The point about Pizzagate is not whether it was common knowledge over here (I only knew about it from following the Trump threads on here and maybe one article in the Guardian) but whether it was common knowledge in the circles Bright moved in.
The answer to that is "almost certainly, 100% yes." So in choosing that name, Bright is in effect thumbing his nose at everyone, doing the narcissist move of openly advertising their own wrong-doing while giggling to themselves at the thought that no-one will realise that's what they're doing.
It's a kind of double bluff: "My mates know Pizzagate was a crazy alt-right conspiracy theory, so when I call myself Dr Pizza they think I'm just making ironic fun of the alt-right and their crazy willingness to believe any old shit, while actually I'm making ironic fun of my lefty dude bro woke mates, because actually I am the very Dr Pizza the alt right are talking about - the only difference is that I, from the top of my ivory tower, know (because I've read Foucault) that I'm not actually doing anything wrong..."