WI Girl,
It's obvious you're beyond frustrated with this, and now suddenly you have dozens of "helpers" throwing even more ideas at you. You're here for help and we're here to help you - it's all good
You need to sift thru ALL the stuff here - maybe write it all down or print it out - whatever works for you... There's a ton of good ideas made so far, and some probably really irrelevant ones. If the stove alone brings on the condition, yet it takes a few days of running to register, and even then it's not something you can definitely make happen, that is a really unique occurrence.
Focus on the main culprit for now - you have a wonky woodstove installation - get a different sweep to do a camera-drop check down your liner, looking for tears. Just cause your Installer says it's fine, isn't reason enough to believe it is, esp w/ the amount of aggravation caused thus far.
You might try a pressurization test yourself - pull the stove out, cap the top of the chimney temporarily, and MacGuyver yourself a compressor / fan / manometer rig to pressure-test (or conversely, leak-check) the flue itself. You would expect that if it was capped at the top, and pressurized at the bottom, that it would build up some serious backpressure (i.e. blow off whatever you tried to cap it with!). Hopefully it's not a crack or tear that only opens when hot...
One other thought is, esp if you've gone to the trouble of making this pressurization contraption, to just run that for awhile - moving basement air up and out of the chimney. Yeah you have an OAK, and you can even duct that into the MacGuyver Box I'm discussing - whatever you want to do - just simulate the overall air / exhaust motion of the stove while eliminating its own CO output as a contributor. Chances are the CO detectors stay at zero, and you will at least have a really good idea where your CO is coming from...
I would consider outdoor-vented (i.e. out the top of a flue) or outdoor-generated (i.e your idling cars, which inexplicably keep getting mentioned) CO sources to be almost completely incapable of being a problem here,
unless there is some inlet drawing air back into the basement immediately adjacent to the CO-producing outlet.
Your paired stacks crammed into the single flue? They meet this criteria.
Again - good luck and we are all hoping for the best here!