Hmmmm yes, no, kind of. The thing is, it looks like a turn of phrase created by the writer. It's not a common expression in English. In one sense, it has a similar meaning to "If you don't grease the wheels the cart won't go" because Baba paid off the city officials, even though they did nothing. I guess if he hadn't paid them, they would have done something to sabotage the project. However, oil for moustaches is a pretty frivolous thing. Baba paid them a bribe of sorts, and the city officials created a silly excuse for receiving that money. They wanted to make the payment look "legitimate", even though they were clearly getting a kickback for doing nothing... they were neither the wheels nor the cart. So in another sense it wasn't money that was properly spent.