I’m going to drop in again to say that Albucierre’s particular solution in his doctoral thesis was a mathematical closed form corner solution for tractability.
We shouldn’t take the features of this limited corner case as characteristic of the drive approach. Instead, we need to understand that the point of his thesis was to demonstrate cleanly that this particular solution was viable to get around the FTL problem in general relativity.
The thing is that the inertia being zero is implied one of the assumptions of the corner solution. That is, for tractability, Albucierre assumed that the ship would have no initial velocity that it would take into the warp bubble with it.
It would be mathematically messier and would require a computational approach to relax this assumption and allow the ship to have positive initial velocity, but it’s exactly what some of the folks trying to extend the model and reduce the exotic matter requirement have explored.
All to say that the elaboration of Albucierre’s approach seems likely to take it exactly in the direction of some of the distinctions the OP has noticed.
Th most significant difference that remains is that ships at warp are able observe and to receive information from outside their bubble while this seems inconsistent with a bubble in Alcubierre’s model.