I bought this pull-up bar ( https://trugrit-fitness.com/products/tru-grit-pull-up-bar-lite ) since it was on sale, but now I'm stumped on whether I am mounting it safely. Here is how I planned to install it, first my walls are plaster so using a stud finder or magnets was giving me inconsistent reading, so I ended up buying an endoscope camera to locate the studs which explains the holes in the walls which I will patch up later. The red outlines show the studs which I found out are 2x4 and are laying on the 4-inch side, so it only gives me 2 inches of depth to screw the lag screws.
I plan on using a 2x4 (which is outlined in black) that will span across 3 studs at the very top and use 3/8 lag screws to mount the 2x4 to the 3 studs. one more 2x4 that will span across 2 studs below the first one, Then another 2x4 that will span across 2 studs 23 inches below the previous 2x4.
The final image has the pull-up bar bracket color-coded in pink which is how I plan to mount it. Should I be worried about it not being secured? are there modifications that I can make or should I ditch the plan entirely?