Rashi on chumash quotes the din that there is no issue of having chametz in your house on pesach sheni. In fact, b'pashtus there is no issur of eating chametz with the pesach sheni. The mitzvah of eating the korban pesach sheni is fulfilled by eating a k'zayis of meat. Once you eat that one k'zayis, the rest of the korban has to be consumed to avoid nosar, but the mitzvah of achilah is over. In terms of everything eaten after the one k'zayis, there should be no problem of eating it on rye or club, whatever you like, as you are just eating to avoid having leftovers. The one k'zayis eaten to fulfill the mitzvah of achilas ha'korban is a different story. You can't eat it on bread not based on an issur chametz qua chametz, but based on a different din -- the achilas reshus of the bread would be mevateil the achilah mitzvah of the korban. This would be true no matter what you eat that k'zayis with, whether it be bread, a potato, quinoa, or something else.
Based on all this, the Minchas Chinuch does not understand Rashi on our parsha (9:10)
פסח שני, מצה וחמץ עמו בבית, ואין שם יום טוב, ואין אסור חמץ אלא עמו באכילתו.
What does Rashi mean ואין אסור חמץ אלא עמו באכילתו? What's the issur of having chametz with the korban pesach? If Rashi is talking about what's eaten after the first k'zayis, there should be no issur; if he is talking about that k'zayis eaten to fulfill the mitzvas achila, then any food, not just chametz, cannot be eaten with the korban. See Meshech Chochma who tries to offer a hesber.
Rav Shach answers this question with a pshat in Rashi that is so simple you have to wonder how no one else thought of it. The way the M.C. reads the word אלא in Rashi אין אסור חמץ אלא עמו באכילתו is that there is no issur of chametz EXCEPT for chametz eaten with the korban pesach. Rav Shach says that's not pshat. The way to read Rashi is אין אסור חמץ , there is no issur of chamtez with the korban pesach, אלא= RATHER, עמו באכילתו, it may be eaten with the chametz! Rashi means exactly the opposite of how Minchas Chinuch read it.