There will be days which you have no choice but to rough it out and farm Stale Cheese.
This means that you will have to choose a trap combi that has a really bad Cheese Effect, put a piece of cheese on it, and camp out in a really rough part of town.
One suggestion is the Sinister Portal on the Explosive Base, and staying in the Lab or Mousoleum.
What kind of cheese to use? Well, if you're going to let it get stale anyway, choose the cheapest cheese.
If you already have access to the Catacombs, that's the White Cheddar for you (Curds&Whey @ 3pp, Catacombs + Salt @ 2pp, Harbour/Bazaar).
Otherwise, you're stuck with normal Cheddar @ 10pp, from Gnawnia.