So the Handwraps of Mighty Blows double the damage dice, but Dragon Blood Sorcerer claws have two dice, 1d4 slashing and 1d6 elemental. Are both dice doubled in this case? Both dice are inherently part of the dragon claws, they just do different types of damage.
As you invest these embroidered strips of cloth, you must
meditate and slowly wrap them around your hands. These
handwraps have weapon runes etched into them to give your
unarmed attacks the benefits of those runes, making your
unarmed attacks work like magic weapons. For example, +1
striking handwraps of mighty blows would give you a +1 item
bonus to attack rolls with your unarmed attacks and increase
the damage of your unarmed attacks from one weapon die
to two (normally 2d4 instead of 1d4, but if your fists have
a different weapon damage die or you have other unarmed
attacks, use two of that die size instead).
You can upgrade, add, and transfer runes to and
from the handwraps just as you would for a
weapon, and you can attach talismans to the
handwraps. Treat the handwraps as melee
weapons of the brawling group with light Bulk
for these purposes. Property runes apply only
when they would be applicable to the unarmed
attack you’re using. For example, a property that
must be applied to a slashing weapon wouldn’t function when
you attacked with a fist, but you would gain its benefits if you
attacked with a claw or some other slashing unarmed attack.
The elemental damage is explicitly treated as extra damage on top of the claws' base damage:
Vicious claws grow from your fingers. They are finesse unarmed attacks that deal 1d4 slashing damage and 1d6 extra damage of a type determined by the dragon in your bloodline.
In this case, the handwraps of mighty blows only affect the base slashing damage.
The term „extra damage“ is nowhere defined. Is it part
of the damage die or not? It sounds as not but it is
far from clear i would say. A huge issue is if they don‘t
until you reach spell l5 for the bonus casting magic
Weapon on your 2h leaded staff does more dmg,
So i assume it is at least intended both dice get added.
If you had a +1 Sword with a Flaming Property Rune on it I do not think that you would double both the striking and the extra flaming damage that it does when you add one Striking Rune to it. By the same token I do not think in this scenario that we would double the elemental extra damage from Dragon Claws.
Additional statement just to make sure there is no misunderstanding: The Handwraps of Mighty Blows themselves would not double the damage dice. This is an effect caused by having a striking rune applied. The Handwraps are just a mechanism to let you apply runes to unarmed attacks.