There actually is a reason for the name- it’s because Batwoman and Batgirl are actually completely different characters.
Batwoman debuted first, but was a rather silly character. She never had her own story and mostly served as a love interest to Batman. She proved to be unpopular and the character was scrapped for the most part. She would appear occasionally in the 70s and 80s to fight by Batgirl’s side or moon over Batman, but the character was in effect retired.
Years later, Batgirl was introduced. The publishers went with “girl” rather than “woman” in order to distinguish the new character from the old. This Batgirl, Barbara Gordon, was clever and capable. She had her own stories within the Batman Comics, but only had one comic of her own. In 1988 the comic “The Killing Joke” came out and poor Batgirl gets shot and paralyzed. She eventually takes up the name Oracle and becomes the leader of the Superhero team Birds of Prey. In the New 52 series she is healed after 3 years and once more becomes Batgirl, although an older, wiser Batgirl with PTSD. Previous versions had her permanently paralyzed but also permanently Oracle. Her age tends to shift around as new interpretations of the character appear, ranging from her late teens to early 30's. She often toys with the idea of becoming Batwoman, but always decides to keep the name she made for herself as Batgirl. There have also been a slew of other Batgirls that were bestowed the title while Barbara was Oracle (in most cases they actually reported to Oracle and were members of the Birds of Prey) but Barbara is the main one and the one we would expect to see in the movie.
The title of Batwoman was brought out of retirement in 2006. There are two main Batwomen, Kate and Kathy Kane. Despite the names, the characters are not the same. Kathy Kane is the original Batwoman. She has been reimagined a number of times over the years, but is generally portrayed as a wealthy heiress with ties to the circus who is a love interest for Batman.
Kate Kane is the current Batwoman. While this Batwoman is also a wealthy heiress, she most certainly is not a love interest for Batman. Kate is a lesbian and a former West Point Cadet.
Sorry for the overly detailed explanation of the name. I hope it was at least somewhat interesting. Basically it comes down to, Batgirl and Batwoman being distinct characters. Batgirl keeps the girl part of her name to preserve her individual identity. She earned that title and she decides who gets to use it.