I used to own an R100/7 with a Velorex car. My solution was to park the rig on close to level surfaces in first gear, utilizing motor compression to prevent a roll away. It seems to me that a person could rig up a piece of metal flat stock to wedge between the bike and side car frame to actuate the rear brake. Parking in first gear always worked for me.

Thanks. Unfortunately, my state requires a parking brake. Wonder if a bungee around the front brake lever will pass?!

I rode a 1969 Triumph with a Velorex Sidecar for forty years. Never gave it second thought - steep hills: park with a wheel jammed in to the curb AND the bike in low gear. On the flat just leave it in low.
It's interesting that your state has a parking brake policy aimed at sidecars. Outfits are normally so far beyond the legislative collective consciousness as to be invisible.
The Velorex brake (only) is not the place to be looking for installing something. On my outfit, I built an attachment that extended the motorcycle rear brake pivot plate so that I could attach the sidecar brake cable to the same lever. That way, I still had separate brakes - front on the handlebar, rear and sidecar collectively on the rear brake pedal, but no parking brake.
If I were you, I'd start thinking along the lines of an additional pedal or removable over centre toggle that locks the rear brake pedal down. Or -
Your bungee cord around the front brake lever idea might be a good place to start. I would think a little outside the box on the bungee cord, though. Maybe a removable fitment that positively clamps the front brake lever would meet the letter of the law as well as the intent.

I was thinking about something like that but will try this first to see if it passes the inspection.

Perfect! (IMHO)
Good luck with the authorities.