I think you have to take the 5k. The only way it doesn't leave everyone better off and save lives is if you don't actually believe your prior of >99%, in which case update your prior. I don't see how what he does in another room matters. Any reputational effects are overwhelmed by the ability to save thousands of lives.

However, I also don't see how you can cooperate in a true one-time prisoner's dilemma without some form of cheating. The true PD presumes that I don't care at all about the other side of the matrix, so assuming there isn't some hidden reason to prefer co-operation - there are no reputational effects personally or generally, no one can read or reconstruct my mind, etc - why not just cover the other side's payoffs? The payoff looks a lot like this: C -> X, D -> X+1, where X is unknown.

Also, as a fun side observation, this sounds suspiciously like a test designed to figure out which of us actually thinks we're >99% once we take into account the other opinion and which of us is only >99% before we take into account the other opinion. Dr. House might be thinking that if we order 15k of either medicine that one is right often enough that his work here is done. I'd have to assign that p>0.01, as it's actually a less evil option than taking him at face value. But I'm presuming that's not the case and we can trust the scenario as written.