I think being "frostily" polite is acceptable if it's what you can manage; "professionally" polite is better but if you are so focused on BEING polite you may not be able to be better than frosty.
More important, I think, is that your co-worker has been written up for her behavior and your bosses know this. So keep focusing on what you say, not how you say it. If your co-worker complains "Boss, I said 'hi how are you' and she said 'good morning' back and it was rude!" then your boss is going to be unsympathetic. Similarly if she complains "I asked about her family and she said 'I'd prefer to focus on work just now'!"
Focus on "what am I saying that my bullying co-worker could repeat to the boss?"