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Result for the Commoner vs Bishop end-game:

6 Pawns each: 61.7% (+/-0.9%) in favor of Commoner (879 games)
Commoner + 5 vs Bishop + 6: 63.4% (+/- 1.35%) in favor of B + P (729
games)

Again, the extra Pawn swings the result by about 25%. The Commoner seems
nearly 0.5 Pawn stronger than a Bishop. The statistical error is ~6 cP. 

So against a Commoner a Bishop seems less successful than a Knight in the
end-game. This could be because it has more difficulty winning games when
it creates an advantage, due to its color binding: two Pawns are easy to
stop by King + Commoner. if they just position themselves in front of the
Pawns on a color the Bishop cannot reach. This is unbreakable defense even
when the attacker calls both its King and its Bishop to the aid, because a
King cannot approach a Commoner either. (And there is no zugzwang, as you
can do moves with the King / Commoner in front of the Pawn that has no King
support.

So the defense is even easier than with unlike Bishops, because there you
have to worry that the Bishop might not be able to stop a Pawn supported by
its King, so the defending King has to keep opposing the attacking one.
Which it can usually do, as the Bishop can control squares in front of both
Pawns. A Knight can also not stop K+P, but if your King aids it in the
defense, there is nothing stopping the other Pawn, as the Knight usually is
too far away. So a defending Knight cannot easily exploit the color-binding
weakness of the Bishop, but a defending Bishop or Commoner can.

This is another case where the unapproachability of a Commoner makes it a
very strong defender. (The other is in KQKM, which is draw if King and
Commoner can protect each other before the Queen snipes off the Commoner
through a fork.)