Comments on the game itself:
'Lighthouse is not worth it' AKA reading this board. There's nothing spectacular going on. The best engine kind of thing at all is to open feast/lighthouse, grab as many markets and lighthouses as possible, 1 militia on the next $4, and roll up conspirators; eventually you may want crossroads and possibly even scout. But that doesn't look terribly convincing to me. So then militia dominates the board. It's definitely a more powerful card for as straight BM a game as this is than market. The buys market gives you won't be so important, because you're really unlikely to hit, say, $13. So, as militia is the most important card... it's quite reasonable to go after lighthouses, I think. Certainly they at least deserve serious consideration. They don't actually produce much less money than silver in the first place. And potentially you can use markets to pick them up and actually make that buy useful. Then there's smugglers, which is always something to consider, but... I think I'd prioritize militia here. Smugglers will be better early; militia most likely much better later on, but the question is which terminal do you want? And I think it's militia, though analytically you probably want to open with smugglers and pick those militia up later on. And then lighthouse also, in proportion to how heavily your opponent is going militia. Thing is, you can actually get a decent number of terminals in this kind of deck, for a couple reasons. One, if you aren't defending yourself with lighthouse, the militias of your opponent will take care of some terminal collision for you. Two, between extra buys and gains off smuggler, your deck will have more cards. Three, crossroads. But against you, you've got the cantrips presented by market, making your effective deck size smaller.
You buy a feast - for more markets? Which you (rightly) say aren't all that great? And again, militia is a really strong card later on in the game, and your opponent's militias reduce terminal collisions
You buy a smugglers because you're behind? Well, he's going to be provincing pretty soon... this is the point where I DON'T like smugglers that much anymore. I guess they can grab duchies... but you aren't terribly likely to be buying a province as well, if you do. And how many plays of it are you likely to get at this point, especially since A) if you're using it to poach a gold or market or something, you have to wait ANOTHER reshuffle to use that card; B) some of those potential plays of smugglers, you expect to get eaten 'cause he's buying province. I'd have preferred silver....
You say something about frisk buying duchy surprisingly early - well, on $5 exactly (i.e., you not on $6+, where you should still prefer gold a while longer), 6 provinces is generally where you want to start taking your duchies, unless there's some very nice 5s you want instead, and market clearly isn't nice enough. On the other hand, it was close to his reshuffle...
'Duchy Dancing' is something I tend to say when you're buying them over province, a la PPR, which isn't the case here. Here you're just buying duchies.
The last province buy for you... well, what're your other options? Duchy+Estate? In that case, you aren't terribly likely to make it back to $8 all that soon. Perhaps your crossroads make you better suited for a long, drawn-out green war? But also, in the only case that matters, where Frisk gets $10+ with multiple buys, if it were really a concern, he could double duchy in response, and again you'd be on the back foot pretty hard. In fact, you were probably just lost either way, so I'd endorse taking the gamble by buying the province.