I've concluded that it's to difficult (perhaps impossible) to use the SDK for BrowserStack. Maybe I can get the next best thing working.
https://www.browserstack.com/docs/automate/selenium/set-name-and-status-of-test?fw-lang=c-sharp%2Fmbunit#setting-the-test-status
This describes two techniques. Ink presently trying to use REST to update the session (test) status but that doesn't seem to be working.
There's also a JavaScript way which I can try out.