You are missing something. FI is not defined in terms of accessibility of functional states from this sequence. It’s defined in terms of target and search space. The search space is the space of all 100 bp sequences. The target space is the set of all 100 bp sequences that perform the same function, regardless of whether their sequences resemble the one that we happen to be looking at (presumably because it happens to be the one used by some lineage of organisms). By my statement of the scenario, the functional sequences constitute 0.1% of the search space. This specifies the FI of this functional class to be 10 bits, whether or not the sequence in question can reach those other states by single mutations.
Whether this particular sequence can reach any of those other functional sequences by allowed mutations has only the most tenuous relationship to FI, in fact. I didn’t state it explicitly, but I meant that functional sequences are randomly distributed throughout the search space. Given that, there is a reasonable probability that one or two functionally equivalent sequences are single-mutation neighbors of the one we’re examining, but the most probable case is that there are zero. Regardless, the sequence as I have described it will be highly conserved, even though FI is low.