The only rumour that I know is that many say that Charlotte is in fact the mother of Maxime Weygand and gave birth to him in Belgium -although there is another alleged mother, a Polish woman-. According with this story his father was a Belgian officer (Alfred van der Smissen) who had an affair with Charlotte, and she was "sent" to Europe (1866) to give birth in secret for avoiding a scandal. Honestly I don't believe in these stories... I do believe she went to Europe for asking for help to their Empire -her life and Maximilian's were in risk-, French anounced they would leave Mexico and anything would convince them to change. Both knew they needed the French army to keep their throne safe.
I don't think that she was pregnant while she stayed at Miramare, I don't believe that she would dare to visit Napoleon III and his wife in that situation -and considering the time, her pregnancy wold be very advanced and would not be easy for the Empress that other people don't notice it-. She would be trying to hide her inconvenient secret, and many people in France would have seen her.