The problem with this solution is that it will inspire people to do the video game equivalent of counting cards: I know that the bag has 20 items, and I've just had bad luck with 10 pulls in a row, better go fight the boss while my "luck" is up.
Edit: though, this is assuming that you know when the bag empties, but that shouldn't be too hard to detect given enough iterations.
Interesting. I did not think of this, but yes, I think this would create a rift between 'casual' gamers and 'hardcore' gamers. IE those that abuse the fact, that the randomization is hampered for gameplay purposes and those who don't.
Thinking of it, this is a real problem, because if the shuffle bags size is increased to make tracking the state of the bag harder, the randomness should converge against true randomness again, defeating the point.
Edit: though, this is assuming that you know when the bag empties, but that shouldn't be too hard to detect given enough iterations.