"Bar Kaibutsu" for the Kaibutsu that can be found in the Sugar Hole on occasion.
Besides the bar Kaibutsu, fans also have named the Kaibutsu Gabe, Carrie, and James.
"Charlotte", for the pink-haired girl that sits by the fire in the alleyway behind the asylum. The nickname is possibly a Madoka reference.
God Never Said That: lol has never stated the gender of the Kaibutsu. The only "confirmation" of their genders is what little can be taken from their sprites, and even then, the only Kaibutsu with an unambiguous gender is the twintail model.
Wiki Rule: Found here. Note that the wiki is somewhat outdated in places and uses names for worlds/events that aren't the commonly-used fandom names.