We are always suggested using MPS with open boundary conditions in ITensor. So even if it is a Hamiltonian with PBC, we express the eigenstate with MPS with OBC. Although this would require more memory, it's convenient to compute entanglement entropy for intervals extended to the end by SVD only on a single bond.
The measurement of entanglement entropy of an interval inside a chain is similar in three cases. A similar question is discussed here http://itensor.org/support/229/evaluate-block-entanglement-blocks-that-extend-the-lattice?show=230#a230. The main problem is if "l" is very big, it's impossible to do with MPS. But we could calculate the second order Renyi entropy by a sampling method, which is very similar to "perfect sampling" in a paper by Ferris and Vidal.