If I have multiple copies of Snitch installed, do I get to expose an unrezzed ICE for each copy, or does it still cap at one?

You get to expose one per copy. Card text usually only refers to the card itself, not other copies, so "using up" one Snitch's ability won't affect your second one at all and since it doesn't have any other restrictions (like "you may expose the frist unrezzed piece of Ice you approach each run", for example), there's nothing stopping you from using each Snitch for a different piece of ice.

I'm not sure I'd recommend it, since you'd have to pass at least one previously unrezzed piece of ice (presumably by paying to break it) to use the second one's ability and even then it doesn't do too much for you unless you find out you can't break it and multiple copies do hog quite a bit of memory, but it's possible.