Is there any way to increase the threat range of an animal companion from the base value of 20?
The only thing I can find is two types of animal companions that have a special bite. But this is probably worse than just taking a 2- or 3-attack monster.
The animal is going to be druid level 11 which rules out trying to go for the Improved Critical feat.
No, without waiting until a much later level for improved critical you can't do this. At level 10, you can pick up Vital Strike (or level 9 if your GM will allow you to retrain your animal's feats) which is kinda-but-not-really similar--but if you want more than one attack Vital Strike is wasted as feat (a wolf works really well for this Weapon Focus, Improved Natural Attack, and Vital Strike then cast Strong Jaw and it works wonders).
Another option that doesn't really work is that animals can make use of Amulets of Mighty Fists which can be enchanted. However, keen is only applicable for piercing/slashing weapons and an Amulet of Mighty Fists can only be enchanted with enchants that are applicable for bludgeoning weapons.
At this point, it's up to your GM to allow you to have either a custom AoMF or some kind of "animal companion" weapon with keen (though even a single keen weapon will only work for one natural attack).
