Whenever I start the car, the passenger side window goes down?

Is this the base model or does it have modules in each door? Did you do a full system scan, because if this is not the base model and has modules in the front doors, they should show up on a full system scan. The door modules are actually on the CAN Bus network. It doesn't seem like a base model, but I can verify by VIN if you'd like. And the first thing to check is the wiring between the door and body in the rubber flex boot. If you haven't already, I know you're pretty good on checking things thoroughly.
And when it does finally start working is it from the driver's side or pass side switch or both?
One other thing, does it roll all the way down or only about 3 inches?
If this does have door modules, you may find no communication with one or both modules.
Oh, you had the battery disconnected; I wonder if they just need the window relearn procedure.
I saved these files from the other day, or yesterday.

https://www.2carpros.com/articles/electric-window-repair

Can you view any live data from the window modules, these modules have hall effect sensors in them to detect position, there is a TSB on the modules, but take a look at the 2nd and 3rd diagrams in my last post, see if you can clear the window memory and do the relearn procedure. I'll post the TSB if needed
And check there's no water intrusion or corrosion in those module connectors, looks like it's been apart before. Also do check that rubber boot the harness runs through to the body.
The window may be going down due to this high voltage at the sensor. These windows will automatically stop travel if there is any resistance felt by the module, so it may be kind of what is happening here.
Also, service info states the illumination is only LEDs, so I would assume that is only 5 volts as well. Since they are such low current components. Since there is a 12v feed, I would verify the ground using that feed.

Diagrams 4 to 8 are the flow chart for the B185D code, if you would like the other flow charts just let us know. These should help though.