As recommended, I downloaded the Python script and placed it in ~/Dropbox. Unfortunately, while that script accepts the commands, whatever command I enter (including the one I want, status), it says that Dropbox isn't running.
$ ./dropbox.py status
Dropbox isn't running!
I don't know Python, but I can see that it needs the contents of ~/.dropbox/dropbox.pid. That file is empty.
Back to square 1, unfortunately.
Re: Using the command line with a flatpak installation