Would a creature in the Blood Frenzy stance and with the Blood Frenzy Strike apply the Bleed effect to the Storm Kindlers Storm Shape damage?
While you are using Blood Frenzy Style, when you damage an opponent with your unarmed strikes or natural attacks, you deal an extra 1d6 points of bleed damage; this bleed damage does not stack with itself
i would say no for 2 reasons:
1. you do not deal your unarmed strike damage but damage equal to it. the whirlwind ability that this mimic does the same dealing the equal damage in slam but has this to say :
"...A creature in whirlwind form cannot make its normal attacks and does not threaten the area around it...Creatures one or more size categories smaller than the whirlwind might take damage when caught in the whirlwind (generally damage equal to the monster’s slam attack for a creature of its size) and may be lifted into the air"
this is not a normal natural attack but a sort of aoe damage (that only effect creatures of specific size) that is based more on the creature's size.
2. while it is not explicitly saying that it's a polymorph effect (which make the user lose racial abilities which in turn would lock you off of amphibian and the style feats) it seem to be some kind of similar transformation and i would rule that most normal racial abilities might not work in this form. mostly those which depend on the physical nature of the creature.
Yeah, "lethal damage equal to her unarmed strike damage" is absolutely not "your unarmed strike". It only uses the damage dice size, nothing more.