I would have gone for the semi-engine of trying to play Milita + Masq every turn. CH/Beggar might work, but it feels risky, since you need to draw CH and Scavenger at the right times to make it scary enough.
There's no +Buy, and all the cards you want (Fortress, Masquerade, Caravan, Militia) cost less than $5, so you don't need to open Silver here. Accordingly, you should focus entirely on setting up draw. Caravan/Masq > Masq/Silver for the open. Caravans are very important because you want Masq + Militia in hand with 2 actions to get the full benefit.
My game plan here is probably Caravan/Masq opening, get 2nd Masq ASAP for trashing, and pick up some Caravans/Fortress before Militia. Early on I'd even buy Caravan over Gold, because you'll have time to buy Gold later. You don't need Militia right away, it's better long term to pick up other things.
Edit: I think the key point here is that in the log, you spend turns 3 + 4 + 5 + 6 + 8 buying treasures or terminal Silvers. That forces you into a money strategy, because you don't have time to buy the actions you want, but you also have a few more actions than a money strategy would want, so you're getting the worst of both worlds.