Do you gain new fleet condition cards when you start a pivotal battle if you would have qualified for a condition in a normal battle?

No. You only gain fleet condition cards if you declare an attack / defense during a normal round’s strategy phase (see rules page 19). This does not happen (there is no strategy phase) for pivotal battles (see page 24 that discusses what a pivotal battle consists of compared to what a normal round consists of (see last page of the rules)).