Also, would wearing a beard make me look more or less foreign?
one stuck on with glue
If anything, the opposite. Especially if you're wearing a false beard. You could try dressing as a nun.
I don't want to be instantly recognized as a foreigner walking in the street.
Warsaw is a cosmopolitan city, especially in the areas near the metro. It has a low crime rate, so you shouldn't worry.
How should I dress for this purpose?
You don't say what country you're from, but the clothes here come from the same sort of chain stores that are all over the western world.