How can I safely mount a pull-up bar on a suspended stud wall?
Your studs and framing should be fine. You need to screw into the existing studs and odds are that the pull up bar won't do a direct attachment very well connection bigger than the width of the stud. Also, the current framing layout will dictate where you place it.

Your idea of attaching a board to the wall and the pull up bar to it are a good one and what I would suggest even if you hadn't thought of it.

I would use a 2x6 or a 2x8 (the wider 2x8 might look better) and screw it to the studs with (3) #8 x 3-1/2" screws per stud to a minimum of (2) studs. Be sure that there is a stud on each side of each of the two pull up bar attachment points, which will likely require attaching the board to three studs. You might want to extend the board the full width of the opening for a better look.

How far out from the wall you place the bar has a significant affect on the force applied to the board, so keep it to 6" maximum.

Then attach the pull up bar per its instructions