I think the short answer is that it would probably be technically possible, but either through some nontrivial modifications of the DMRG algorithm
(not just small modifications to the current DMRG code in ITensor) or by using a different optimization method like gradient optimization, where it is easier to implement optimizations with nontrivial constraints like that but would likely be slower than DMRG. My guess is that in the end it would be a lot of work and not really worth it.
It is a bit easier to do with infinite MPS algorithms like VUMPS, but I found that there were subtle numerical issues involved with actually getting it working (for example details about which particular matrix factorizations you use when orthogonalizing the effective wavefunction to maintain the symmetry properly). I also only tried it for 1 site unit cells where it is more obvious how to implement that kind of symmetry, I think it would be more subtle for multi-site unit cells.
In the end, you would at most get a factor of 2 speedup, which is ok but I think there are other strategies we are working on for real space parallelization and parallelizing over sums of Hamiltonians which would be more general, easier to implement, easier to maintain, and scale to speedups beyond a factor of 2. We are working on them now so hopefully they will be available to try out relatively soon!