The halacha is that a person over bar-mitzvah who ate to the point of satiation (kdei seviya) is obligated min haTorah to say birchas hamazon and a katan who only has a chiyuv derabbanan cannot be motzi such a person in bentching. R’ Akiva Eiger (O.C. 186) has a fascinating safeik regarding the case of a katan who ate k’dei seviya for dinner (as my son regularly does : ) just before the night of his bar mitzvah, bentched, and then while he still had that satiated feeling it turned dark and he was bar mitzvah – does he have to repeat birchas hamazon? R’ Akiva Eiger writes that his son-in-law told him the Chochmas Adam raised the same safeik regarding an onein who ate and still felt satiated after the burial of the meis. As R’ Akiva Eiger notes, these two issues are not exactly the same. By the case of onein, the person is a bar chiyuva, but because oseik bamitzvah patur min hamitzvah, he is exempt from bentching while he has the onein status. By the case of katan, the person was not a bar chiyuva at all when he ate.
It seems from R’ Akiva Eiger that the issue here is how to define the role of seviya. Is it the act of eating which triggers the obligation of bentching, and satiation is just a condition (a tnai) in the definition of eating, or is it the state of being satiated itself which obligates bentching, provided that state occurs through an act of eating.
Another possible way to view the safeik here is whether a katan, though he has no chiyuv, can have a kiyum mitzvah. R’ Soloveitchik brought a proof to this issue from the case of a katan who becomes bar mitzvah in between Pesach rishon and Pesach sheni. The Rambam paskens (K.P 5:7) that as long as a katan was counted in a group that brought the first korban pesach, he need not bring a pesach sheni. Even though the katan had no chiyuv to participate in the first korban pesach, he still gains a kiyum mitzvah from doing do which exempts him from pesach sheni (this proof is cited by R’ Reichman in Reshimos Shiuirim, Mes. Sukkah; however, see GR’Ch al haRambam there). Perhaps by birchas hamazon as well, though the katan had no chiyuv to bentch, his birchas hamazon counts as a kiyum mitzvah min haTorah. This safeik is relevant to many other areas as well, including the famous case of a katan who becomes bar mitzvah during sefira.