The current way of using Bitwarden in Firefox private mode is really tedious, if you aren’t logged in you have to open a normal window, login then use the context menu to auto-fill.
I tested quite a few other password managers and none of them have any issues with Firefox private mode. I read the issue on the Firefox bug tracker however by the looks of things using getBackgroundPage isn’t the only way to access content, other password managers clearly have figured out an alternative.