Was her name really Bathsheba? Her name is BATHsheba and she is famous for her bath on the rooftop...

Is this just some goofy English translation going on, or just a strange coincidence?
The apparent wordplay of Bathsheba bathing in the story of David and Bathsheba in 2 Samuel 11 is purely an artifact of English translation.

In the original Hebrew, the "Bath" in "Bathsheba" has no connection at all with the word translated "bathing" in some translations of 2 Samuel 11:2.

The Hebrew name בַּת־שֶׁבַע (bath-sheba) is a compound word composed of בַּת (bath), "daughter (of)" and שֶׁבַע sheba "seven" or, as often interpreted, שָׁבַע (shabah), "oath." So the name means "daughter of seven" or perhaps more likely, "daughter of an oath."

The Hebrew word for "bathing" or "washing" in 2 Samuel 11:2 is רָחַץ (rachats).

So in the original Hebrew there is no pun or wordplay involved in Bathsheba bathing.

Meanwhile, the English word "bath" comes from Germanic, not Hebrew, roots. It has no connection with the biblical story of Bathsheba. See "bath" at the Online Etymology Dictionary.