So, upon seeing a new one, you’ll have to resolve it. Only then would you be able to tell what other URLs it’s equivalent to.
One way to gain anonymity there is to do that from a proxy, but such proxies would be detectable from the amount of pages they request from Facebook.
It also looks like they already thought about replays of URLs. For me, https://www.facebook.com/VICE/posts/pfbid02XdVziPTwhmPU9XzBq... currently says:
“It looks like you were misusing this feature by going too fast. You’ve been temporarily blocked from using it.
If you think that this doesn't go against our Community Standards, let us know.”
So, chances are they also thought about users exchanging URLs (e.g. by having each running instance of Firefox read Facebook URLs for other instances). It is possible that (a part of) your Facebook user ID also is encoded in each URL.
Or encode some versioning scheme, and keep trying various versions until one comes up with a valid link. If we can think of these things in seconds, so can the engineers at FB.
One alternative, which would require significant effort and investment but would be a brilliant way to outsmart Facebook's crap, would be to accumulate pbfids in a common pool such that, if a given pbfid points to post X, fetch a different random pbfid that points to post X. If the initial pbfid is not recognised, add it to the pool once the post is determined, either as a new alternative for a known post, or as a novel entry.
Of course, FB would hate it and would either try to expire old pbfids (and risk breaking "legitimate" links) or use legal threats, which would require them to openly admit that they don't give a shit about people's privacy preferences.