Plumbing below grade question

I have posted before about our toliet troubles. We have had two professional plumbers out and both said they are not sure what to do. Onw suggested busting out walls and replumbing. We don't want to do that. I think we would just not use the bathroom instead but it sure would be convienent I'm hoping that you guys can help give some suggestions. Here's what we have. The house built in 1942. Bathroom installed sometime in the 1970's to our best guess. It is a ranch with basement. Well water and septic system. We have had our septic inspected, pumped and all the lines from the house to the septic were replace approx 1 year ago with PVC. The basement is concrete floor with all solid concrete walls and 3/4 finished. The basement bathroom has a toliet, sink and shower stall. It seems we can wash our hands without a problem but if we use the shower the water backs up out of the pipe in the floor by the sink. We have a brand new Kohler Wellworth toliet in that bathroom. There is a stack pipe with a cleanout outside the bathroom wall. We can not flush the toliet period. The only way to get it to flush is to uncap the clean out on the otherside of the wall and then flush. It flushes perfectly then (as long as we leave it uncapped). So we are considering a different toliet or one of the Qwik Jon System. The only problem that I see is if we use something that has an extra box or something is finding space to fit it. The toliet is mounted to the ground and since in concrete and between the door and shower it could not be moved. The wall behind and to the sides are solid concrete walls. Hence our problem with putting something that needs to be piped into the toliet. We have approx 1 foot between the wall and toliet.
I hope this is enough information that maybe you can give some suggestions.

toilet

You are describing a situation where the toilet is "air bound". If so, then it could be because of many possible reasons, but one constant would be that the pipe leaving the bathroom, and past the cleanout opening has standing water in it, or is a closed system. One possibility to go along with that is if the bathroom was installed with an air admittance valve instead of a vent throught the roof.

my 2 cents, and i don't have the experiance of many here

you say ,when you open the cleanout cap it finally flushes.
chances are the basement toilet was done many years after original const.
it may not be vented at all! it may be vented, but not properly. you could have a birds nest in the vent. my guess is,it wasn't plumbed right.
i also wonder, how sharp those 2 plumbers were. i think a sewer camera is in order. hang in there ,smarter folks are on the way, tomorrow ,on this site

Some more information

We paid to have the line from the septic and clean through the house as far as they could tell augged(I think that is how it is spelled). Never really found anything and the toliets were not flushing when the folks left. Yet they insisted they corrected the problem. Anyway, they suggested changing out the toliets which we changed out 2 of the 3. the basement and one on the upstairs. The oldest one is the one that works the best but it still only swirls most of the time but will flush if we keep flushing it (once we get the problem fixed we will change that one out as well as it is an original toliet). My husband found a vent line in the attic but because of concrete walls and solid ceilings (some of them are concrete as well!!!!!) we can not trace the pipes. He feels that the basement may not be vented at all. The thing is the original owner's (owned 50+ years)family insists that it worked fine as far as they could recall. The owner (owned 3 years) prior to us said they never used it so they didn't know anything about it. We do believe it is a venting problem but we want the least evasive way possible.
I don't know if this helps at all but the sinks and bathtub upstairs drain just fine.
Oh yea, we are going to snake the vent we found but we don't believe it has any leaves or bird's nest in it since it is only vented into the attic. Not sure if we should bring it out in the roof but we would wait until spring if we had to do that. The air admittance valve may be something to look into as it is only vented into the attic.
Hope this info helps.

the vent must go through the roof and depending on where along the roof line it goes will determine how high above the roof it must go. That vent (assuming it is not clogged) is a direct line from your septic tank into the house - the gasses could find their way into your house and be a major health and comfort issue. It is not likely any vent for the toilet goes through the concrete walls, if the walls are unfinished by a stud wall and you don't see a pipe going up from the bathroom in the basement, it probably is not vented. To work properly, this must be changed.