In loop quantum gravity, the volume operator for a given region is not a diffeomorphism invariant. But classically we know that volume is a scalar quantity under a diffeomorphism even if we take the full manifold or any region.
Why would you think that volume is not diffeo-invariant?
If the region moves along with the diffemorphisms (aka passive diffeomorphisms), the volume is invariant. In LQG, the volume is also invariant (if you disagree, please explain why).
If the region doesn't move with diffeomorphisms (aka active diffeomorphisms), the volume changes. That is because the region here is only a region in the coordinate space, which doesn't have a well-defined notion of volume. The same is also true in LQG.
LQG is really not different from General Relavitity when it comes to diffeomorphism invariance.