Just a thought here and I'm no expert at all. Hopefully Claude, Jay or others will correct if I'm in error.
On all my bikes I like to set the preload on steering stem bearings so that they're snug, with just a slight resistance to turning. I lift the front and after repacking stem bearings will tighten the steering stem nut etc. I usually find that bikes are set up so that with wheel straight ahead just a touch on the bars causes wheel to flop all the way over to the side. I prefer to setup so that it's tight enough that it takes a nudge to keep the fork turning side to side, just a slight preload resistance. That if you move bars a couple of inches the front wheel stays there unless you give another very gentle push. This of course with properly lubed stem bearings.
I'm unfamiliar with your stock steering damper and how the stem is set for preload. You might want to investigate what kind of preload is going on with damper backed all the way off and VW Damper disconnected. Establish what kind of resistance is present at zero damping setting, and also at progressively more dampened settings, to see where you're really at. From there you may be able to add just a wee bit of preload when in the zero setting and eliminate unwanted play in steering stem, but not to cause binding.
While you're at it you can spin front wheel a few times to see what the wheel & tire balance looks like. See if the same part of wheel always ends up at bottom. Also check lateral side play in the whole wheel assembly.
With the chair's shock/spring disconnected & wheel off the ground I'd check to see what kind of play may be present in sidecar swingarm assembly? You could check to see what amount of damping is offered by sidecar's shock & spring too, that it's firm enough?
As the experts said, sounds like you're pretty close right now, maybe just some minor issues. Happy adventures to you!