How can I farm for a specific type of heirloom?

Solution 1:

You are correct in your guess. Each dungeon has specific chances for the heirlooms you find to be each type. Each dungeon also has one type that is 'common' there, which means you'll receive this type more often, though it is not guaranteed. This type is:

type        dungeon
busts       ruins
crests      cove
deeds       weald
portraits   warrens

The second thing you can do is making sure you prefer going to 'longer' missions. With more tiles and rooms in a dungeon, there's a larger number of loot curios that can hand out heirlooms.

Another thing you can do is to bring a high amount of provisions to a dungeon; with more provisions you can get better rewards from curios. The exact types you need to maximize heirloom rewards depend on which dungeon you are running. I constructed the below table from the relevant wiki page. Also see this question

dungeon        bring
cove           skeleton key, shovel
ruins          skeleton key, medicinal herbs
warrens        skeleton key, holy water 
weald          skeleton key, shovel, antivenom
courtyard      shovel, the blood, bandage, torch

If you are playing with districts, then you can build the cartographer's camp. It ups loot chance, which can increase the amount of heirlooms you get. Depending on how many upgrades you're planning to get the large investment of 300 crests may or may not be worth it.

Note that bringing an antiquarian will not help you here.

Finally, loot rewards are bigger in darkness. By doing a dark run or by snuffing out your torch prior to looting a curio you can increase the amount of heirlooms you receive.