Let's say that I gave my pet T-rex two extra heads, each with a bite attack.
Would those extra heads get the double Strength bonus to damage from powerful bite? Because it says "bite damage," and that seems pretty general to me.
If I'm right, primal T-rexes are terrifying.
I don't see why not. The ability doesn't specify what head benefits from the extra str damage, so it should work.
And the only relevant restriction would be "only one natural attack per appendage", which you don't break since it has several heads/bite attacks.
Remember- if it has 1-2 natural weapons, it can get a single extra 'iterative' style attack (ie- a bite at BAB-5).
So it gets three bites in before doing haste and such.