Actually the BPS view is that brain and mind are different and interact.
My take on this is they vacillate on this view, and retreat to the position I stated when pushed. They do not necessarily believe it though.
I think they reason along the lines you discuss a lot of the time too, if not pushed. Their position is not consistent.
However, they most definitely are the new breed of dualists on body-mind theory. They have repeatedly claimed that mind and body are the same or equivalent, but you cannot pin them down on details. I used the line "brain and mind are the same, but mind is more important " because it highlights two conflicting views that they operate in. I think its mostly a smokescreen so they can focus on mind.
They sometimes comment that the mind and brain are both biological, but never take it seriously. Instead they take the view that mind is paramount, brain secondary, in a lot of discussion. The tactic seems to be to acknowledge all sides, and then claim only their view is important. Its a variant on the balanced reporting tactic.
Stated views and views taken in discussing their hypotheses and models are not always the same. They rely heavily on the vagueness of their claims. They switch claims and explanations on the fly.
Finally they also discuss social aspects, and then relegate them to offhand comments so they can focus on mind. Only social factors that impact mind seem to be important. Somehow they rarely discuss how social factors impact biology, an example of which would be meal preparation and eating habits and H. pylori spread.