When I was computing E.H.M. ground states for V > U > 0, which nearly approaches a product state as V >>> U, the entanglement goes down.
This sounds to us as a good thing, but in fact with DMRG it is a bad thing because it takes an extremely large number of fast sweeps in order to converge. In essence, the small number of nonzero entries in the SVD leads to poor communication of information across the lattice with each sweep.
For instance, I believe I needed to run hundreds of sweeps to get the energy to settle down, even for small system sizes, and I observed the entanglement entropy changed very slowly with each sweep.
In order to test this, I'd recommend changing your sweeps parameters so that you stay at a single maxdim for several hundred sweeps and wait until the energy settles down before increasing maxdim.
On the other hand, if your boundary conditions are incorrect (for instance, anti-periodic instead of periodic) you can also see wild changes in sweep to sweep energy; I experienced this error one time due to a mistake in my definition of the Hamiltonian.
If you figure it out please post back here as I'm interested to know what you find, as I am continuing to use these methods.