I think you start going for Duke, when you've finished building your engine. Here it is when the grand markets run out. The smugglers are really helpful for a Duke game. I think your opponent used one to get a grand market, duchy and a duke.
You need to compete for duchies if your engine can't buy all of the provinces. Here your engine has no card draw, so will start chocking pretty hard on provinces. You need to compete for duchies. 6VP dukes for $5 should beat anyone going for provinces.