I feel like I already know what the answer to this is going to be (and I'm fairly certain it's already been discussed, but I could not find the thread), buttttt....
Interdictor: triggers when a ship activated (even itself)
PQ7s: triggers when the ship it is equipped to activates
So the interdictor activates and uses PQ7s. They exhaust. Interdictor title triggers and readies it. Since the same timing is still being applied, you exhaust PQ7s.
I know the RRG states that an effect can only be used once per trigger for multiple keywords (words to that effect), but should "exhaust" cards be treated differently? I'm not saying have a new rule made for them, but have the current rule interpreted specifically for this type of instance. The "instance" being that the exhausting of the card is what triggers the card effect (which is basically how it already works, though I know exhausting the card is treated as an effect), versus the actual timing. Thus, the trigger for the effect happens twice in the case of the interdictor, and the effect doesn't happen twice under the same timing.
And I know I'm about to get shot down, but the interdictor could really use some love. It's had a Regionals season almost as bad as the Falcons blowing that 22 point lead, and letting it PQ7 twice in the same activation wouldn't be game breaking...