My gut leans towards the cost and VP being balanced. In two player you need a 7-3 of a pile to get value, and it's well within someone's means to block that. It's almost unheard of for a player to get a 3-7 split when he was doing all he could to stop a particular stack from emptying.
At a 6-4 split of a pile, it's still interesting because you can buy Province alternatives, forcing your opponent into a longer game. Meaning if you split the Oases 6-4 you could be in a good position. So that's cool.
You can do some decent playtesting solo, I've tried it before; there's definitely things you can pick out and learn. I set up "bots" against eachother, comparing one strategy directly to another head to head, and making sure the strategy that ought to win does. (for instance, if nothing on the board synergizes with your fan card and it still can beat BM/Embassy then there is probably something overpowered about your card.)
The nice thing about your card is that, being a Duke variant, it's tough for it to be overly weak. The card will definitely change games, you have to mirror Minion in a game where normally you would try an alternate strategy, you have to contest the Lab stack even if you want to play Big Money, etc. So I would skip to making sure it's not too strong. I'd suggest pitting a "bot" that buys nothing but one Smithy, Labs, and money against a bot the also prioritizes Labs, to block the Matrimony strategy, but ends his Lab chains in a Steward. The Smithy-Lab bot goes for Matrimony then Duchies or Provinces as appropriate, while the Lab-Steward bot should go for Provinces (he's trying to end the game, his deck is slimmer and accomodates green more poorly).
I would think you want the Steward bot to be the one that wins in this situation. If it can't, it would appear to me that the card is a little too dominant in two player, and forces the long game too easily. Steward's trashing goes with the Lab strategy better than the no-trash BM approach, so if the Steward player appropriately cuts off the Lab pile at a 5-5 or 6-4 split then he should win.