Would a firebender still have killed her mother, or an earthbender on account of the war?

I’d say yes, because I sort of see Asami feeling the same way a lot of other Yu Dao citizens felt; in that, they may be Fire Nation citizens technically, but they felt like Earth Kingdom citizens because some families had been there for generations. There could have, of course, been some firebenders who were fiercely loyal to Ozai and resented those that felt that way and saw it as a betrayal. Asami’s mother could have been in the crossfire of one of those conflicts.