Take another look at the sentence:

Even though Frances had already ate, she stopped at the food court and bought a slice of pizza. She wanted an opportunity to smile at the cute young man who ran the register.

The first choice, had ate, is incorrect. [Already is an adverb and not officially part of the verb]. This verb requires the past participle eaten because of the auxiliary verb had.

Verb Simple Present Simple Past Past Participle
eat eat(s) ate eaten

Bought, the verb you chose, is perfectly correct. Before you continue, review the rules for irregular verbs.

Try again to become a champion!

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