When you think of how many times Donna Troy's origin was retconned, explained, re-explained, you could almost have a little sympathy for
Bette Kane! She is, though perhaps not exact, the re-visioned "Betty" Kane who was the first Bat-girl.
But there are so many disparate and desparate attempts to explain or remake her, that it almost begs for at least one coherent story to explain her (keeping in mind that Wonder Girl had several, so why shouldn't Bette get one?)
"The Secret Origin of Bette Kane, Flamebird" (told you I was a geek)
Okay her father is either the "unnamed son" and brother of Jacob, Martha et all, or Phillip. I think whoever it is, is Jacob's twin brother
as twins run in families, Jacob being the father of twins too. It also runs, genetically perhaps, that one twin may more likely be homosexual and I propose that Jacob's twin brother is gay. Here's where I dive in and yeah, rather Soap-Opera-y!
In that older generation, even a wealthier family expected a gay son to father children and leave heirs, but maybe this brother really didn't want to do that and somehow contrived to let his brother consummate the marriage. Maybe alcohol and partying was involved. Drunken twins and who's the daddy!
So maybe, Jacob is really Bette's father... When I think of his rather extreme grief over Bette's cutting by the Hook its almost like he saw another daughter going to die.
So that's what I think. That Bette is Jacob's secret daughter and also Kate's unknown half-sister. Or maybe subconsciously realized on Kate's part, hence a bit of that protectiveness.
Then there's the Flamebird thing. The best reference to why or how she has that name is an obscure Superman story from the early 2000s where Superman and Lois return to Kandor (I think ) and also replay the time-honored roles of Nightwing and Flamebird. He indicates to Lois that he told both Dick and Bette of the legend.
When would that have happened? My personal canon and best shot is that it occurred after the events of the classic Silver age "Prisoners of Three Worlds", which was revisited by Morrison in more recent years. Batman, Robin, Batwoman (the first obviously) and "Bat-girl" are exposed to substances that cause a pretty wild adventure. Bruce even declares he loved Kathy Kane. Robin and Bat-girl seem to travel to another planet. etc.
What if Bette had the hardest time recovering from the substance, being the youngest of them. Who would Bruce turn to if the factors involved extra terrestrial agents? Superman of course.
So Superman attempts to cure/cleanse Bette and meanwhile she and Robin learn the Kandorian/Kryptonian legends from him. Although super-hypnotised Bette retains a part of this experience. Maybe Superman's cure imparts something that boosts her abilities or creates the affinity for the Flamebird. Years later . .
Tah-Dah!