The Mishna in Pesachim already deals with the problem of how to be mafrish challah when you are baking fresh matzah (as they did in the times of Chazal) on Y"T. R' Shternbruch in Moadim u'Zmanin (vol 4 #285) deals with a similar case where a person finds himself on erev pesach she'chal b'shabbos with matzah which he forgot to be mafrish challah on. He cites an interesting solution suggested by the Maharil Diskin: give some of the matzah to a 12 year old muflah ha'samuch l'ish for his own use and allow the child to be mafrish challah from his batch for all the matzah on Shabbos.
How can we encourage or even allow the child to be m'chalel shabbos? A few years ago we dealt with the gemara that says a katan can eat from the korban pesach even without minuy, the "reservation" that a gadol would have to make to be able to eat from the korban. The Rishonim ask: if a gadol would not be allowed to eat from the korban, then wouldnt feeding it to a katan be like giving him maachalos assuros? Ran (Nedarim 36) explains that minuy is only required for someone who has a chiyuv to eat korban pesach. Since a child has no real chiyuv, no minuy is necessary. Tos (Pesachim 88) disagrees and writes that a katan in theory does require minuy to be able to eat from the korban, but for the sake of chinuch in the mitzah of achilas korban pesach we dispense with that requirement and allow a katan to eat. This Tos is the foundation for Maharil Diskin's argument: for the sake of chinuch in a kiyum mitzvah we are even allowed to cause the katan to violate what otherwise would be assur.
(It's worth noting that the Minchas Chinuch (end of mitzvah 7) writes that Tos should not be taken at face value as meaning a katan has carte blance to violate issurim so long as it is l'shem doing a mitzvah for chinuch. The whole point of chinuch, writes the M.C., is to train the katan to do a mitzvah in the way he would have to do it as a gadol. A gadol does not do issurim to accomplish a mitzvah, so neither can a katan. Tos has to be taken in context. The reason minuy failed in the gemara's case is because by definition, a katan is excluded from minuy -- it is his age which is the barrier. Had the katan been over bar mitzvah, the minuy would have worked. He is doing the mitzvah exactly as he would have to as a gadol. See Meishiv Davar I:41 )
R' Shternbruch raises many, many arguments against this chiddush. Tos is speaking about giving korban pesach to the katan on pesach -- that is the only possible time for chinuch in that mitzvah. In the case of hafrashas challah on shabbos, there is no reason to be mechanech a child in the mitzvah of hafrashah then and there.
One argument he raises that is relevant to most of us this year is attack on the idea that a katan can be mafrish for someone else on Shabbos. R' Akiva Eiger (159) raises the question of whether a kinyan made during the week can be chal on Shabbos. Why should doing business be different than any other melacha? I can start my washing machine on Friday so it finishes a load on shabbos, I can set a timer on Friday to turn on a lamp on Shabbos, so why can't I close a business deal with a clause so the sale takes effect on Shabbos? The Avnei Nezer explains that the washer will finish no matter what, even if the owner of the clothes dies; your timer on your lamp will continue to operate even if you leave the house, sell the lamp, and run away; however, a business transaction does not work that way -- there has to be a buyer and a seller, the transaction is inextricably linked to the actors involved. You cannot separate the action that occurs on Shabbos from the person. If you empowered your Rabbi to sell your chametz, one of the issues he would have to deal with is this R' Akiva Eiger. Can the mechiras chametz be done on Friday and take effect on Shabbos, or must the sale be completed before Shabbos with a loophole to exclude the chametz needed for Shabbos? Getting back to the Maharil Diskin, R' Shternbruch writes that the hafrashas challah done by the katan works through zechiya, i.e. the katan is acting on behalf of the owner of the challah with his approval. If a random person were to take challah from dough without the owner's consent and permission, it would be meaningless. So here too, like in a business transaction, the concept of ownership, or owner's consent, means the person is inextricably linked with the act. It is not at all clear that this should be permitted.