How much times the strength modifier does a chained barbarian add when making a bite using Animal Fury?

Question

The assumption is bite is the only attack the barbarian makes during the turn, and the barbarian has no other natural attacks. I seem to get three different answers on this:

The barbarian adds 0.5x the strength modifier. This is because the rules on Animal Fury state that "If the bite hits, it deals 1d4 points of damage (assuming the barbarian is Medium; 1d3 points of damage if Small) plus half the barbarian’s Strength modifier."

The barbarian adds 1x the strength modifier. This is because the 19 STR cannibal from the gamemastery guide has its bite damage listed as 1d4+4, which is consistent with this.

The barbarian adds 1.5x the strength modifier. This is because a bite is a primary natural attack, and the natural attack rules state that "If a creature has only one natural attack, it is always made using the creature’s full base attack bonus and adds 1-1/2 times the creature’s Strength bonus on damage rolls."

So which one is it?

Answer 1

The barbarian adds 0.5x the strength modifier.

Generalized rules are used unless there is a specific case that overwrites it. This happens when certain criteria are met. In this instance, adding 1.5x str mod to a single natural attack is the norm, unless it is overwritten by a specific case. The barbarian's Animal Fury is an example of a specific case that would take precedence over the normal rule.

As for why the Gamemaster Guide's Cannibal gets away using 1d4 + 4 damage on the single bite attack, I'm not sure. The other damage types it does are consistent with adding strength mod from the rules (two handed weapon attack doing +1.5x str, single handed attack +1.0x str, and bite doing +0.5x str)

Answer 2

It is your second option... probably

The phrase about damage supplements previous one about determining bite's attack bonus during full attack. So it talks about your Strenght bonus to that bite attack under certain circumstance only - you make a full attack combining manufactured and natural weapons. For that specific situation, rules say

Creatures with natural attacks and attacks made with weapons can use both as part of a full attack action (...). Such creatures attack with their weapons normally but treat all of their available natural attacks as secondary attacks during that attack, regardless of the attack’s original type.

The only thing Animal Fury really does, it gives you a bite attack while you rage. When he isn't making full attack your character uses the folloving rule

Primary attacks are made using the creature’s full base attack bonus and add the creature’s full Strength bonus on damage rolls[...] If a creature has only one natural attack, it is always made using the creature’s full base attack bonus and adds 1-1/2 times the creature’s Strength bonus on damage rolls. This increase does not apply if the creature has multiple attacks but only takes one.

This is actually a really tricky point which of the above sentanses (bolded or not) should apply to a charactr able to perform attacks with manufactured weapons, but who has only a single natural attack. Pathfinder bestiary isn't realy helpful here because of a way creature's statistics are given. But Pathfinder is a direct inheritor of dnd-3.5e, where statblocks were a bit different and can provide some help here. Here are Minotaur and Bralani entries, which both indicate 1x Strength modifier on a creature's single natural attack used not as part of a full attack. I searched more, and was unable to find any monster, who use manufactured weapons and gains 1.5 Strength on a natural attack. Your cannibal example also match this.