Club World is simply a marketing term for long haul international Business Class. Don't let the name they call it trip you up.
Still not clear on your intent, beyond getting into some form of actual business class. Is someone (employer?) PAYING for a ticket for your wife? A coach/economy ticket? or are you personally on the hook for getting your wife a seat on a plane to Italy (either using all miles or a combination of a paid ticket and miles). There is a difference, and it's important. As you have seen, upgrading a paid coach ticket results in a different "landing spot" when comparing AA to BA.
Does it HAVE to be on BA? Your location is "west coast". Where exactly? Routing may make a big difference. Can your wife fly from Los Angeles? You could get on the new 77W (777-300) which has a very nice business class hard product.