Both moves have other qualifiers than "start of session, if it feels right". Moonlighting explicitly needs downtime, but doesn't care about sessions. Skipping a week in-session is the kind of things the Operator loves, but if less than a day has passed since last session, nothing happens. Everyone does stuff "on-screen", but the Operator is special because he, by default, keeps doing stuff off-screen, whenever that is.
Wealth needs a secure position or whatever, and can admittedly become a bit weird, especially if you let the barter from the surplus accumulate. I'd say that "having surplus at hand for the session" and "surplus: 2-barter (you're personal share)" combines to make "you can grab up to 2-barter from just around your compound without anyone caring, but if you don't use it during the session, it doesn't automatically go into your personal hoard." And if you play that in mind, the hardholder can roll for Wealth pretty much regardless of time passed since last time, as long as his hold is secure and his rule unchallenged.