Increasing the distance from the wall of my pull up bar
For just 1.5 inch more space off the wall, sure, another layer of 2x4 could work fine. It's not necessary to pull the existing horizontal boards off the wall; you could just build up another layer in front of those. Thinking about how the forces might be in play I guess orienting the second layer vertically could be a bit better.
If you wanted to hang the arrangement from the ceiling.. what if you rotate the brackets so that they attach to the bottom side of the I-joists? If that puts the bar a little too low you could install some 2x blocking between the webs of the floor joists and fix the brackets to that instead.
On the other hand, if brackets mounted to the I-joist flange puts the bar too high above the floor then you'd have to find a way to bring it down lower. Lots of strategies to do that.. some kind of lumber hanging down from the joists. Might be a patch of plywood fixed to the side of two floor joists, and a piece of 2x4 at the lower edge of the plywood to give a secure surface for mounting the pull-up bar brackets.