"Adoshem" and "Elokeinu" In Teaching Kids Brachot

So we were lighting the Chanukkah candles tonight, and I was trying to walk Junior through saying the brachos together with me. So I got to Baruch Ata Adonai, and he stopped and got upset, because I was saying it wrong. His teachers had taught him the "proper" way to say it. I should have said "Adoshem," and "Elokeinu."

Now, whether one could or should use that particular made-up Divine name ("Adoshem," as opposed to "Hashem") in non-ritual contexts is up for dispute. Kitzur Shulchan Aruch (6:3) is against it, based on Taz (Orach Chaim 621:2). See here, in Pitchei Halachot But apparently, the Rav would say Adoshem in certain contexts (citing a pasuk, if I heard correctly. And so did others.

Even so, when teaching children, the Mishnah Berurah, Orach Chaim 215:14, rules (again, see Pitchei Halachot) that even the teacher may use the shem Hashem, namely Adonai Eloheinu, and not a kinui, to ensure that the children learn it correctly. (And I would say, based on his language, that this is how he understands the Shulchan Aruch.) Not so fast, though! The Aruch Hashulchan, Orach Chaim 215:2, states that while the children may say it normally, even though they are only practicing, the teacher should substitute "Hashem" (he does not say Adoshem, though I can see why they use it, since it has the same number of syllables) and "Elokeinu." And that even though there are those who are mattir, it is proper to distance oneself from this. (Unless he is only saying this latter about the second clause, about using Shem Hashem in the course of expounding in public.)

So Juniors teachers certainly have what to rely on, in this chumra. But then, they should make sure that the children, when practicing, make use of the actual proper Divine names. And this, even Aruch Hashulchan allows. Unless there are other sources in play. I did only very surface research into this.

However, at the very least, they should make sure the kids know that when actually making the brachos, they should use Adonai and Eloheinu.

Note: Do not take this as paskened halacha. Consult your local Orthodox rabbi.