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You can actually figure this out by doing what it says at the beginning of the core rulebook. It works surprisingly well.
I don't have my core rulebook on me, so I can't quote it verbatim, but the rule goes like this:
If you have some situation where two multipliers would be added to the same roll, the multipliers are not multiplied together:
---For example, spirited charge says you deal x3 damage when charging from the back of a mount with a lance. The lance has a x3 crit multiplier. When you crit with a lance while charging you do NOT deal (3x3=9) x9 damage.---
Instead, take the second multiplier (x3) and subtract 1 from it (x3-1=x2) You then ADD that multiplier to the previous multiplier (3+2=5) you deal x5 damage (Still awesome!)
So, while charging with a bite attack and dragon style, you deal x1.5 your strength, and x1.5 your strength. Subtract 1 from the second multiplier (1.5-1=0.5) and ADD it to the previous multiplier (1.5x0.5=2) There you go, double damage.
Please note, that the above was for calculating your STRENGTH bonus to damage, not your damage overall. So if you crit while charging bite with dragon style, you still do x2 damage. That would make the formula:
(bite damage die+(2x str mod)+additional mods to damage)x2