Banjirou will appear only on the original facility, since you have White 2, that means Banjirou will appear only in White Forest's Treehollow.
He won't appear if you change the area, he will be replaced by a Veteran instead. If you beat the Veteran instead of Banjirou, that counts as beating a Veteran, not beating Banjirou, so you won't get a shiny, until you swap the City back into the Forest and challenge Banjirou.
That sucks, I really wanted the shiny Gible instead of the Dratini