So the most straightforward answer to your question is that you cannot efficiently measure this particular quantity from an MPS representation of the ground state. Sorry.
One way to see this is to think of your sites not as 3*N dimensional sites, but as separate sites for each of your sublevels. So this would be a representation of your problem having a unit cell, with the first site being sublevel -1, the second being sublevel 0, the third being sublevel +1, the fourth being sublevel -1, etc.
From this point of view, the entanglement you are asking for is like the entanglement of every third site, starting with site 2, relative to all the other sites. But as you probably know, the only entanglement "cut" which is efficient to obtain from an MPS is a left-right bipartition.
Ok that's the straightforward answer. The more complicated answer is that you may be able to obtain this quantity if you "swap" the sites or reorder them. So if you apply swap gates to move all of the sublevel 0 sites to the right of the sublevel +1 and -1 sites then it would be a left-right bipartition. But this is unlikely to scale well, so it's probably inefficient.
Another idea is that you could perform sampling to obtain the second Renyi entropy of the sublevel 0 site entanglement.