While I was creating the characters in Guilty Gear, I had a spot for a cute character. I thought it would be too boring if the character was just cut… - Daisuke Ishiwatari

After the events of Bridget's story in Arcade more, she self-identifies as a woman.
So, as to whether “he” or “she” would be the correct pronoun for Bridget, the answer would be “she.”
As mentioned in her character profile, Bridget was born as the younger twin son and then raised as a girl by her parents to protect her from a village superstition.
Despite their intentions to protect Bridget, it pained her parents to do so as they felt they were forcing her to live a certain way.
Bridget, realizing this, attempted to bring wealth to the village while behaving like a man, thus overturning the village superstition, as a way to free her parents from their guilt.
As a result, the superstition faded, and nothing remained to restrict how both Bridget and her parents lived.
After this, Bridget tries living as a man, but it doesn’t feel right.
This is where the Arcade Mode story begins. After her exchanges with Goldlewis and Ky, Bridget faces parts of herself she has tried to ignore, and makes a big decision for herself.
I hope that all of you will watch over her path after her courageous choice to stay true to her own feelings.

Bridget: You’re not taking Roger away from me!
Bridget: I’m... I’m a boy. These girl-clothes are... It’s a long story.
Bridget: Thank you very much! It’s all thanks to you! I’ve got to toughen up! Y’know, since I’m a boy and all... Heh heh...
Bridget: Hey! I-I’m not a “kid,” okay?!

両性具有なんです。まぁもう人間を超越した存在になっちゃってますしね。俺もだけど。
They're androgynous. In fact, they've transcended human existence. Just like me.