Let's say I've got a rigger jumped into a stirrup interfaced guard dog. That rigger wants to make a bite attack.

Is that Gunnery or Unarmed combat?

I think it's technically Gunnery, but I vaguely recall something about biodrone natural weapons being unarmed for riggers... but I may be mixing that up with biodrones being able to use natural weapons via the animal's unarmed instead of a pilot + autosoft.

Oh.
As far as I can recall, biodrone rules always have been kind of unclear.
I'd go with unarmed combat. That is set on riggers piloting a biodrone having to use their own skills (and I hardly see vehicle skills apply at all) (see "Augmentation", biodrone game information) and the fact that if it were a normal humanoid drone and the rigger were to slap someone with its mechanical arm, he'd also have to use his own unarmed skill (see "Arsenal").
My german sourcebook quotes would probably not be of help right now

Wouldn't it be an exotic melee weapon? Like the bite rules?

No, bites gained through SURGE or other qualities are Exotic Weapons, but those that are Powers (like a critter's bite) are Unarmed Attacks.