Just wanted to check… are we talking about the BW addon not working in a private window because Firefox by default does not allow addons to access private windows unless you specifically check the box during addon installation (or the radio button on the addon settings)?
If the “Run in Private Windows” option is not checked, then the following message appears when you are in a private window and click the BW dropdown “Unfortunately this window is not available in private mode for this browser.”
If its a bigger issue with the API, please disregard, it wasn’t clear to me from the OP that this was the case.