There are some posts related to this but they are all 5 months old before the FAQ ruling that shooting makes you ineligible to shoot.

If my opponent has a secondary mission such as Cleanse or Deploy Teleport Homers and declares a shooting target instead of performing a secondary action, can they perform the secondary action if the shooting target uses Sigil of Exigence to become ineligible? This is assuming there are no other valid shooting targets.

I think the answer is no, and that since the unit declared to shoot there are no longer eligible to shoot (and therefore ineligible to perform a secondary action), even though they ended up with no eligible shooting target.