Installing a Tension Chin Up Bar - Can I Use Shims to Minimize the Pressure on the Drywall?
As you have found out there needs to solid structure in the wall behind the drywall, I.E. a stud. Typically these are mounted in doorways that have solid structure and i am not sure where you are mounting yours.
A stud  in a wall is only 1 1/2&quot; ( &quot; = inch ) wide so the bar would need to be perfectly center, but even if centered the whole width of the mounting plate of the bar may not be on the stud and drywall is relatively soft so the movement of you on the bar could cause the bar to weaken the drywall.
Studs are usually spaced 16&quot; on center, so A better approach would be to use a thick board to span across some structure, two studs. (or what ever structure you are dealing with, adjust the mounting board accordingly)
A 2&quot;x 6&quot; board 18&quot; or 20&quot; in length would span two studs. You need to find the center of each stud so you can screw the board into them.If you do not have a stud finder then you can find YouTube videos to learn alternative methods.  Mark the center of each with tape. place the board on the wall so it spans the studs and transfer the mark on the tape to the board.
Pre-drill your board with drill bit just smaller then the thickness of the screw, place the board on the wall, position the board so the holes line up with the marked tape on the wall, use a level. (You can eyeball it if you need or use a tape measure from the ceiling)  The board opposite it will need to be in the exact same height/position. Use a drill bit slightly less then half the thickness of the screws  to punch through the holes in the board, this will mark the wall where you need to   pre-drill the wall/studs.
Pre-drill the drywall/studs to 1 3/4&quot;, put tape on you drill bit so you know when to stop drilling. You can skip the pre-drilling of the wall if you have an impact driver and you use torx head screws. (Phillips head screws suck) Mount your boards to the walls and mount your bar to the boards.
EDIT: After thinking about this a bit it occurred to me that your shim idea May (?) be sufficient Without the need for fasteners to screw it into the studs. Use a large thick square board, pinch it between the ends of the bar and the wall. You do  want to make sure that you are centered over a stud otherwise you could punch an 8 inch hole into the wall instead of the smaller one  you already did.