thank you for the great work on the insights pages of your websites. I am having difficulty understanding why the gemarah wishes to prove from the fact that the tanna of our mishnah cannot hold like neither r' yehudah or like r' meir that that undermines the validity of chananyah's approach. he is just as much a tanna as they and it shouldn't be difficult for us to set the mishnah up like chanayah. furthermore why then is the gemarah interested in at least trying to reconcile the mishnah with r' yehudah part way? if the intent of our gemarah is to determine which of the rabanan or chananyah our mishnah goes like this would seem tangential to this discussion. it would logically appear that the only way to prove it is by asking something along the lines of the first question from r' chelbo but by proving that two other tannaim don't hold of chananyah that isn't more powerful than the rabanan who also don't hold of chananyah. the gemarah went to great lenghts to show us how the mishnah can fit in with chanayah's opinion so it is obviously valid - the fact that the baraisa would then argue on our mishnah should not be a cause of great concern.
Your question is a very difficult one. Indeed, it appears that that is actually the answer of the Gemara -- that the Mishnah does not agree completely with either Rebbi Yehudah or Rebbi Meir, but it is the opinion of a different Tana altogether. That Tana is Chananya. However, it is difficult to understand what the Gemara thought initially.
Perhaps we could suggest that since we do not find that Chananya mentions the argument between Beis Hillel and Beis Shamai, it could be that since Rebbi Yehudah and Rebbi Meir do mention Beis Hillel and Beis Shamai, and Chananya does not, it makes sense to say that Chananya is a different opinion altogether and argues on both Rebbi Yehudah and Rebbi Meir, and if so, the Halachah is not like his opinion.
The Gemara concludes that Chananya, when he stated his opinion, was following the opinion of Beis Hillel.
The RITVA writes a similar approach.