How do I mount this pullup bar?
Your approach is sound. I wouldn't bother toeing screws in from top/bottom. I would drill through both the stud and the filler board and attach with nuts/bolts/washers (as opposed to threading a screw into the stud). Two bolts above the bar and two below.

Alternatively you can attach horizontal 2x4's across your studs with recessed lag bolts and then mount the pull-up bar to those. Since the 2x4's will be attached to the studs the same way the pull-up bar would have been attached you won't lose much connective strength.