I've never heard of an object with a natural armor bonus, but I suppose it's possible. If it does apply, then it would improve the armor class of the mirror . . . but its armor class is already very bad, much worse than a typical familiar, so the FCB is unlikely to help in a significant fashion. When a witch gets a new familiar, the FCB would apply to the new one, so it'd be the same for the mirror witch--again, assuming you could apply the FCB in the first place.
Whether it works or not, I would never take this FCB option.
The mirror witch doesn't have a Mirror Familiar, it has a Patron Mirror that works for the Mirror Witch sort of like a familiar works for other witches, but it is not a familiar.
That FCB wouldn't do anything for a mirror witch.
At 1st level, a mirror witch creates a direct link to the power of her patron. She chooses a patron as normal, but instead of gaining a familiar that serves as the conduit to her patron, she uses a mirror (worth at least 10 gp) to speak to a fragment of her patron’s power. This fragment teaches her magic and acts as a counselor.
The mirror is essentially an immobile familiar. The mirror witch can activate it with a minute-long ritual that calls the patron fragment to manifest itself through the mirror. While this connection is open, she can prepare spells, add new spells to the mirror, and use the mirror’s special abilities. In place of the boost granted by an ordinary familiar, the mirror grants the mirror witch a +2 bonus on Diplomacy checks regardless of whether the mirror is nearby. The mirror can’t be used for abilities that require the presence of a familiar, such as the blight hex.
If the patron mirror is destroyed, the mirror witch can replace it 1 day later through a special ritual, attuning a new mirror to a different fragment of her patron. The mirror witch must purchase or procure a new, suitable mirror and spend 500 gp per witch level she has. The ritual takes 8 hours to complete, and when it is finished, the mirror witch establishes a connection with a new fragment of her patron and loses access to spells stored in the previous mirror, as normal for a lost witch’s familiar.
As the mirror witch progresses in level, the mirror’s defenses increase, and the connection between it and the mirror witch becomes stronger, allowing the patron to manifest more of its personality—see the Patron Mirror section below.
This ability replaces the witch’s familiar.
Some text implies that it isn't a familiar. (Example: ... instead of gaining a familiar that serves as the conduit to her patron....)
Some text implies that it is a familiar. (Example: In place of the boost granted by an ordinary familiar..)
Objects are easier to hit than creatures because they don’t usually move, but many are tough enough to shrug off some damage from each blow. An object’s Armor Class is equal to 10 + its size modifier (see Table: Size and Armor Class of Objects) + its Dexterity modifier. An inanimate object has not only a Dexterity of 0 (–5 penalty to AC), but also an additional –2 penalty to its AC. Furthermore, if you take a full-round action to line up a shot, you get an automatic hit with a melee weapon and a +5 bonus on attack rolls with a ranged weapon.
Based on this, I would argue that it is not possible for any inanimate object to benefit from AC based on this, but there could be additional information somewhere that would show they do.