Other than possibly alchemist, what mistakes did I make?
First, I think Alchemist is definitely NO mistake here. King's Court, Vineyards, Grand Market, Bank. You don't win this board with BM-Margrave.
Focussing on the endgame, given we are at turn 14, you are behind 3 Provinces. The probability is very high that your opponent can repeat the strong turn, in which case you lose if you buy 2 or more Provinces. So I don't think grabbing 3 Provinces is the right move. The Problem is also that you have at most 4 buys, while your opponent has a lot more. So everything else also will let you lose, maybe it was not that bad to take the gamble after all...
With more buys, you should have gone Duchies and/or Vineyards, buy say all Duchies and force a longer game. But as it went, I think it's mostly first player advantage, and I don't see how you can reliably turn this into a win.