I study warp drive metrics and their associated energy densities to great extents at my research programs, and I will tell you that if you're talking about an Alcubierre/Van de Broeck/Natario-style warp drive, definitely super certainly surely not.
These metrics rely on the mass-energy density being roughly spherical around the ship in most cases, although they can be made to have toroidal geometries of mass-energy density instead. That doesn't mean that there can be regions where the energy density is small. The "shell" itself will be made of unobtainium fluid, which has energy densities on the order of trillions or quadrillions of quintillions of solar masses per cubic meter; if you safely modify the metric in such a way that a "hole" appears that would allow you to walk into the warp drive's area, it would need to be done in a smooth way, and without making the edge ultra-sharp (and thus making spacetime ultra-bent in that area), even the "hole" will be filled with solar masses' worth of energy.
Theoretically, there could be a configuration of negative mass-energy that would allow for a gap in the shell with energy density below $1000c^2\approx8.98\cdot10^{16}\,\mathrm{J}\,\mathrm{m}^{-3}\equiv1000\,\mathrm{kg}\,\mathrm{m}^{-3}$ (which, in the case of a non-flowing fluid/liquid, would make traversing the gap roughly like swimming through water), but that would be extremely difficult. Essentially, the issue is that the mass-energy densities of the warp drives are so astronomically high that even some distance away from the edge of the bubble, where the energy density is dozens of orders of magnitude lower than inside the shell, the mass density is still extremely high by human standards, and the fluid making up the bubble would be a virtually-impenetrable solid - in some cases even the density of the "gap" would be comparable to the density of an atomic nucleus, or equivalently a neutron star.
So unless you can walk through neutron stars like they were air, or if you can give me a lower-energy warp drive metric, this isn't going to work. If you can do the latter, please stop asking questions on WB:SE and prepare a paper to publish immediately.
But it's no fun not being able to use warp drives and bigger-on-the-inside rooms and spacetime engineering of that sort!
Yeah, it's not. My solution is just to say that the civilization that has access to that tech (maybe humanity, maybe not) at some point figured out a very specific pattern that allowed them to do that kind of stuff at much lower energies. This is specifically what I study - low-energy warp drive solutions - and based on my work I theorize that it is at least sort of maybe possible in the future. With that in mind, as a writer, I just handwave "they figured out how to do it with a 1 GW fusion reactor instead of a Universe-evaporating vacuum-decay engine", and my readers tend to go with it. After all, unless you're writing for a science crowd (which in your case I don't think you are), exact scientific realism can sometimes be given a nudge in the name of something that drives the story and helps flesh out the world.