By what mechanism is a flat region of spacetime imparting motion to this ship? Let's look at the phenomena in GR it's based on.
First is an expansion of spacetime. In the expansion of the Universe, as relatively near points in space obtains space like separations, the discontinuity in simultaneity, as defined by each point relative to each other, increases accordingly. Yet you never actually get anywhere, only farther away from everything else. If you try to emulate this locally, you can't maintain a continual expansion, else it'll have 1 of 2 effects. The first is the location of the expansion doesn't stay local, and must eventually expand to effectively include the entire Universe, pushing you away from everything. The second is that the gravitational field strength must continually increase around the flat region.
Ok, let's not have a continual expansion, but merely create this flat region inside a constant field. It's not even generally true that 2 inertial observers with no relative motion between them will share clock rates. Consider a large uniform massive hollow sphere. As you approach this sphere you clocks will slow all the way to the surface. Go inside this sphere and spacetime is flat. Yet the clock rate inside this flat region of spactime will remain slowed to the same rate as the surface with gravity. So flat spacetime itself doesn't get you out of the time dilation effects.
So this drive creates an expansion and a depression to balance clock rates inside and outside. But there is no constant expansion 'rate', beyond what Hubble law imposes. Only a stable expanded region. You can move around inside this flat region, like in the hollow sphere, but the motion of the expanded regions are not going to impart any motion to you without an expansion 'rate'. You must supply the motion yourself. If your clock rate matches that outside the region, you only get normal relativistic time dilation effects keeping up with the bubble. If you lower the gravitational depth of this flat region, like in the sphere, and get there much faster locally, but that means your clock is far slower than clocks outside that region.
I read the paper, and it moves from expansion to balancing constant fields, as if it's going to accomplish the same thing. Then supposes somehow moving this field isn't going to bring you into contact with the fields gradients, the only thing that can impart local acceleration to you. It seems to understand the need to balance the fields to maintain clock rates, but accelerates coordinate systems as if a coordinate system is a physical thing, it's not. It's called background independence. Changing coordinate transforms of the surrounding space means nothing to your relative velocity to anything in the Universe, assuming your clock rate remained constant relative to the outside, except the fields created by the transform, which then runs you over.
