You know what you should tell her...

its none of her business if someone has a mental illness or not....and how should she react to a crisis such as that at work? Offer her services as a friend and coworker to the distressed coworker....not treat her like "someone with bipolar" but someone who is upset and needs comforting or support....

honestly dreamrunner...i dont think she would be a person to tell....she sounds like someone who label you in a second...and tell people that you are bipolar and explain to them how she feels that you need to be treated....thats when sentences like "Well she is bipolar, so we should tell her the only reason she is crying is because of her mental illness..and its really not that bad"

And yes i have been in sort of the same prediciment....its hard but you will find someone....someone who wont judge you....if they support you and dont have to know you are bipolar to support you....than i think they are worthy enough
thats my opinion...good luck