How to attach a bar in a doorframe
Reline your jamb with another material that you can screw into. The pressure you will subject the door frame to is really needing something screwed into it. To just use pressure has the strong possibility of deforming or worse, cracking your jamb, depends on where the shims are placed behind the jamb, but you cant see them, unless you take it apart, not a good idea, totally counterproductive.

There is more to sketch regarding the assembly how it fits together at the bottom. The idea is, it may be complicated to initially assemble/make, but that time spent now will pay off on how easy it is to assemble and take down.

I see it being 4 pieces, a plywood bottom with pulley mechanism attached, a hinged door head mounted to one of the side jamb liners with the pull up bar and tension bands attached and the ropes to link it all together.