I am Ashkenazi. I was eating with a Sephardi and a katan (under bar mitzva). Were we supposed to do a zimun? Answer:
We start with the only source I found on the topic, which provides practical (intuitively logical) guidance without explanation. Then we will provide the background and critique the ruling. V’zot Haberacha (p. 132), basing himself on communication with Rav Auerbach, Rav Eliyahu, and Rav Scheinberg, gives the following compromise. Zimun can be done, led only by the Sephardi. If there are nine and a katan, the Ashkenazi should answer the zimun without Hashem’s Name.
The gemara (Berachot 48a) accepts Rav Nachman’s opinion that a child counts toward zimun if he knows to whom we bentch. So rule the Rif, Rambam (Berachot 5:7), and Shulchan Aruch (Orach Chayim 199:10), whether for a zimun of three or of ten. (Only one of the quorum can be a katan – Mishna Berura 199:25). You seem aware that this is the minhag among Sephardim. On the hand, the Rosh (Berachot 7:20) cites and accepts the implication of the Yerushalmi that only youngsters with two pubic hairs (the sign of basic physical maturity) count toward zimun; the Rama (OC 199:10) and Ashkenazi practice accept the Rosh. Indeed, then, we have a conflict between minhagim. In a mixed group, whose minhag should "win out"?
One thing to investigate is: how important is each side’s ruling to them? According to the Sephardi ruling, is there an obligation to do zimun or is it only optional? There is an opinion in the gemara (Berachot 45) that there is an optional zimun when two eat together. However, I did not find any indication in the poskim that a zimun including a child is deficient in any way. Therefore, the indications are that the Sephardi member of the group should feel a need to do zimun, for when a zimun is called for, it is forbidden to bentch without it (Shulchan Aruch, OC 193:1).
Is it forbidden (i.e., for an Ashkenazi) to do zimun which is not required? From the discussion of an optional zimun for two, we see that according to the opinion that it is not optional (which we accept), it is forbidden. What is problematic about a zimun’s words? The Shita Mekubetzet (Berachot 45b) says that even a zimun of three is a mini davar shebekedusha (something that requires ten), making it forbidden without its (reduced) quorum even without uttering Hashem’s Name. The Pnei Yehoshua (Berachot 45b) says that it is a disgrace to call out to a single counterpart to praise Hashem, as it is not sufficiently significant.
Is it better to err on the side of recitation or omission? Note that the stronger indications are that a katan counts for a zimun. The implication of the Mordechai (Berachot 172) and Rama (Darchei Moshe, OC 199:4) is that we refrain from zimun in order to be on the safe side. Thus, out of doubt, one would not do a zimun. So how can the poskim we cited expect an Ashkenazi to risk an improper zimun in order to afford his Sephardi friend a "less important" opportunity?
Apparently, the poskim reason that the main problem with an unwarranted zimun falls on the initiator (i.e., the mezamen). Once the Sephardi asks the Ashkenazi to praise Hashem, answering is less of a problem and actually it is a problem to refuse to praise Him. Using Hashem’s Name makes it beracha-like, which is problematic even without initiating. We can appreciate, then, why the Ashkenazi should neither say the Name as part of ten nor lead the zimun.
This resembles the situation of a Sephardi who calls out Barchu at the end of Monday-Thursday davening in an Ashkenazi shul. The minhag is to answer, probably because once it is reasonable to call it out, how can one not respond. In some ways our case is worse. Regarding Barchu, even when there is not a need, the ten provide a tzura (form) of a proper recitation. In contrast, if a child does not count, the zimun is innately lacking.
In the final analysis, there is logic both to and against the poskim we cited. In the absence of contrary sources or compelling logic, we obviously accept their ruling.