It is not strongly established historically that Imam Husayn (A) had a daughter named Fatimah who stayed in Medina when he was travelling due to illness. For instance, al-Shaykh al-Mufid only mentions one Fatimah who was born to Imam Husayn (A) who was reported to have been in Karbala.
Of course, it is certainly possible that this happened especially because historians do not usually not mention everything about young children, and I don't personally feel there is any harm in discussing it in majalis despite the absence of sources from the angle that it could have happened, or it is said to have happened.
However, Bihar al-Anwar does include a report saying that Imam Husayn (A) had a daughter named Fatimah (and she was al-Sughra, the younger) who stayed in Medina. It does not say why she stayed. However it says that a crow came to her with the blood of Imam Husayn (A) (vol 45 / p 171).
Anyway, history is a complicated and murky subject. We should not be surprised that there is little information about some things; rather, we should be surprised that after so many years we have so much!
God knows best.