In Insights, you wrote-
>>However, this alone does not answer our question. In Shevuos, the person definitely became Tamei on one of his trips into the Mikdash. In our case, though, it is possible that the Tumtum or Androginus might not be Tamei at all!<<
How can you say the person is definitely tame when he cleansed himself before he entered the second time?
Although the person in the Gemara in Shevuos cleansed himself before going into the Mikdash a second time, the sequence of events is: he went down one path, went to the Mikdash, cleansed, went down path #2, and then went into the Mikdash. Being that one path is definitely Tamei, he definitely went into the Mikdash at least once while Tamei.