I need some clarification on the card "Fear of Discovery".

In the third quest of the black riders saga, Flight from the ford, you have to build a burden deck.

When travelling to "the last Bridge", you must reveal the to card of the burden deck and resolve it.

I reveal "Fear of Discovery". It says to exhaust the ring-bearer, or reveal an additional encounter card.

Now, my question: Can I choose the exhaust option if the ring-bearer is already exhausted?

In cards that say "either do X or do Y" you must choose one of the options that you can fully resolve, if possible. In this case, you cannot exhaust Frodo because he is already exhausted, so you must choose the other option.