Feel free to write something, but I would focus on the factual things. Ie the sign was broken and she jumped into your conversation. Mention the tone, but understand that such things are hard for them to address after the fact.
I would actually dwell on the tone.
To me it's the only really egregious thing.
Sometimes they use the wrong sign (I still want to know--why the word "broken"?). Happens.
Sometimes a coworkers steps in to help when the conversation isn't going well.
But snotty tone of voice? That's factual. That's material. That's IMPORTANT in customer service.
As I said, I think you should write if it will help YOU let go of this.
You know how we often suggest writing out a huge, long letter full of all the anger and frustration--and then throwing it out? That the act of writing it is often very cathartic?
Well, consider that you get the same benefits, and maybe more, by mailing the letter (esp. if you can write it without vitriol).
I wouldn't dwell on your teariness, etc., but focus on the snotty tone. And feel free to mention your emotional upset and the resulting condescention. That her reaction to upsetting you was to further act as though it was your fault and not hers.
I think we should all have in our toolbox the quiet phrase, "Would you like to say that again with a different tone of voice?" spoken as a genuine question.