There are several non-exclusive ways this can be handled. "Only When Following Deity's Wishes" is a typical Limitation for clerical spells. It's usually -1/2, but that can be adjusted. "Must Stay In Deity's Favor" might be more limiting, maybe -3/4 or even -1. Balancing this might be tricky, as you don't want the character to receive a large Limitation for something that might not come up, or, conversely, a small Limitation for something that routinely impairs the character.
The second way is, as you suspected, a Physical Complication. This scales better than you might think. A character with only a few spells will be Barely Impaired (+0) if he loses them. A character that relies on spells might be Slightly (+5) or even Greatly Impaired (+10) when they lose all of them. Don't worry too much about scaling the Complication to the power level of the campaign.
The third way is to have the character take a Psychological Complication involving their faith. It makes sense that a fickle deity would only grant spells to those with the greatest faith.