Stefan Birkner's answer gave me the direction that I need to be able to solve this. I have posted the code that I used to solve this below.
Solved tests: Birkner's version (recommended)
Solved tests: piped version
Changed source:
WHY: What happens is, with Birkner's library, you can only ever read as much input as you instantiate with the rule originally. If you want to iteratively write to the endpoint, you can do this with a pipe hack, but it doesn't make much of a difference, you can't write to the input over the pipe while the function is actually running, so you might as well use Birkner's version, his @Rule is more concise.
Explanation: In both the pipe hack and with Birkner's code, in the client being tested, multiple calls to create any object that reads from System.in will cause a blocking problem where, once the first object has opened a connection to the Pipe or to System.in, others can not. I don't know why this exactly is for Birkner's code, but with the Pipe I think that it's because you can only open 1 stream to the object-ever. Notice that if you call close on the first buffered reader, and then try to reopen System.in in your client code after having called it from the test, then the second attempt to open will fail because the pipe on the writer's side has been closed as well.
Solution: Easy way to solve this, and probably not the best because it requires modifying the source of the actual project, but not in a horrendous way (yet). So instead of having in the source of the actual project multiple BufferedReader creations, create a buffered reader, and pass the same reader reference around or make it a private variable of the class. Remember that if you have to declare it static that you should not initialize it in a static context because if you do, when the tests run, System.setIn will get called AFTER the reader has been initialized in your client. So it will poll on all readLine/whatever calls, just as it will if you try to create multiple objects from System.in.
Notice that to have your reads segregated between calls from your reader, in this case BufferedReader, you can use newlines to segregate them in the original setup. This way, it returns what you want in each call in the client being tested.