How to keep Animal Companions from falling behind PCs?

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Though I think a bunch of level 3 slots (Greater Magic Fang) all day is a far smaller cost than a single 4th level slot and an action every fight (Superior Magic Fang). You've got enough higher level slots that you can make do with fewer 3rd level ones. Lesser Rod of Chain Spell works but only if you first Ocular Spell or Reach Spell GMF as it's range touch.

Depends how many natural weapons we're talking about, how many fights you get into per day on average, and how often you get into fights without time to prepare first. But you're right that often GMF is going to be the better choice.

Good idea with the Lesser Rod of Chain Spell. If that's not an option and the number of natural weapons in question is not ridiculously high, for a little more gold you could get a couple of Pearls of Power if you don't want to use up too many of your 3rd-level spell slots.