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What happens if a beast that is stationary fails a frenzy check?
Does it even have to make one?

A beast that frenzies immediately activates and attacks the nearest enemy/friendly model.

Beasts thst are stationary cannot activate.

IMO, you take the threshold check as normal, and if you fail, frenzy occurs as per the rules, but the model doesn't actually activate. Then you leave/remove any fury points as you wish and carry on

From the faq.

Replace the last paragraph with the following: A player cannot intentionally choose a path for a frenzying model’s charge that would cause the charge to fail. Because a frenzied warbeast activates in the Control Phase, it cannot activate during the Activation Phase that turn. During the frenzy activation, a frenzied warbeast can only take its movement and action as described above. It cannot be forced, use its animus, use abilities, etc. If a frenzied warbeast cannot immediately activate, it loses its activation and is no longer frenzied, and you can remove any number of fury points from it.

Therefore, if you can not activate, you do not activate and then you remove fury.