Well, he might wonder why she never got introduced by Metratron at least.
The thing is, Frank's intuitiveness is kind of the only thing I've got going for him to make him helpful, so ideally I don't want him taken in, but I absolutely don't want to ruin your plot. If he stays away from her he's not going to have any basis for suspicion, really.
Oh my! Well, fortunately Eileen isn't the jealous type. :x