I see what you mean about cutting to the centre of the stud, to give a fixing for the new board. Yes, you could cut with a Stanley knife. Not really hard work.
I would begin by taking out the old PB up to the edge of the studs first, using a hacksaw blade in a Stanley knife type handle (not a frame). That just leaves a strip of PB attached to the stud. At least then you get a better idea of exactly where to cut with the knife, and can probably easily see/feel best way to remove the strip.