The answer is yes under additional conditions on norm and conorm of $A$ and $\alpha$. Transformation $A$ should be dominated by $T$. A precise and short way to put it is the following:
The map $(x,v)\mapsto(Tx, Av)$ is partially hyperbolic.
Exponent $\alpha$ must be sufficiently close to 1.
Then Walkden's paper "Solutions to the twisted cocycle equation over hyperbolic systems" proves that $g$ is also $\alpha$-Holder. His result is more general.