You obviously didn't trust her though. You snooped through her purse and hired a PI? Why would you not just say "hey, while putting money in you purse, I saw this and I'm sorry. What's going on with that ID?" Then asking her if it's ok for you to look into it? If you had talked to her this wouldn't be an issue.

You were worried she violated your trust, but you ABSOLUTELY violated hers. There are dozens of reasons she might have two ID cards, ranging from a marriage to trying to return it to a friend. You assumed the worst and went to an extreme. That's why YTA. Obviously you should want to protect your kids, but you are an adult and should be capable of having adult conversations.

If it were me, I would quit. I wouldn't want to work somewhere I was obviously not trusted and where my bosses refused to discuss concerns with me.