I stand corrected, it does work! You are a genius!

The key is you have to make sure the app recognized that you are logged in before you start your search.

At first, the only 2 options I saw were Regular or Award Travel and Corporate Travel. I picked Regular or Award Travel and did not see the BfB rate. Then I did Corporate travel just to see the Corporate rate. It asks me to login, which i did. Then I see the corporate rate (but it's locked to 1 person). Went back to Regular or Award Travel and still same non-BfB results.

Then, I clicked on My United, which I think trigger a login into my account. Now when I go to select Book A Flight, I see 3 options: Regular or Award Travel, Corporate rate, and Break for Business rate!

Unfortunately for what i was looking for, the corporate rate was much better than the BfB rate, but that will vary and at least we have an option now.

Also unfortunately, when using the app, you don't have the option to Pay in person with United rep.

In summary, to get multicity to work with BfB rates, go to the app and do something on the app that requires a login, like My United. Then click on Book a Flight.