I would probably go Masquerade, Fortress, Militia, Scavenger (and caravan can never really hurt). But that's not a particularly strong combo, so that's probably why Beggar/Countin House was a thing.
And cellar helps to get Counting House more/faster.
Glad I'm not the only one who had that thought. Would your buy order differ from mine?
Mine was different. I wanna go for setting up the very strong discard attack as fast as possible. I think you bought way too many stop cards that don't help set up the Masquerade-Fortress-Militia deck. You end the game with very little draw, which makes the possibility of hitting with the discard attack pretty unlikely. My deck ended up looking something like 2 Fortresses, 1 Masquerade, 1 Gold, 4 Caravans, 2 Silvers, 1 Copper, 1 Militia + VP cards. Since there is no +Buy, getting more money than 8$ doesn't do much if you're drawing your deck. The attack is so powerful against most strategies that it's still worth it to delay greening in favor of setting it fully up, hence I think I bought Caravan over Province once. I also think Scavenger is pretty superfluous, if you wanna play it you just have to get more Fortresses, and it doesn't do that much in your deck once you're drawing it. Caravan is however a key card here in my view, since it is the only consistent draw (Masquerade and Adventurer also draw, but bleeeh). I opened Caravan/Masquerade, then delayed getting silvers and only prioritised getting my draw up, trashing and playing Militia as often as possible. Maybe Counting House/Beggar could be good on a 5/2? Probably still loses to the engine.