So the way i'm interpreting your question is as follows: The way that the MO school teaches Hebrew isn't consistent, it seems to take a little from here, a little from there, and kind of plops it all together. And so you want to teach a consistent Hebrew pronunciation.
So if your goal is to teach a consistent pronunciation, you basically have 5 to choose from.
The Ashkenazi pronunciation. You are probably familiar with this one. א and ע only serve the purpose as vowel holders, ח and כ share the same sound, as do ק and כּ. The patach and kamatz are distinguished as are the ת and תּ.
The Israeli pronunciation. The Israeli pronunciation is interesting because originally the council in charge of it tried to enforce the pronunciation to be modern Sephardic, in that one should distinguish between ח and כ and א and ע. Unfortunately this never spread to the general public. You are probably well aware of how Israeli Hebrew pronounces the letters, so i won't go into that. But i will say that Israeli Hebrew generally ignores the pronunciation the "sheva/shewa" na in lots of instances. A few examples would be: when the sheva/shewa follows the qamats vowel and other long vowels. When there is a sheva/shewa underneath a letter that has a dagesh in it, a good example would be a recent blessing said during Hanukkah.
The second to last word should properly be pronounced "ba-ze-man" rather than Baz-man. Ashkenazi Hebrew often fails to follow these sheva/sehwa rules as well.
So be aware that if you choose this pronunciation, that built into the system is certain bad grammatical issues.
The Modern Sephardic pronunciation. This is a stripped down amalgamation of several pronunciations and is the most widely used amongst Sephardim today. They distinguish between ח and כ. And between א and ע. They also distinguish between כ and כּ, and ב and בּ. They do not distinguish between patach and qamatz.
The Old Mizrahi pronunciation. This pronunciation crops up amongst certain community members that recently immigrated to Israel or to other countries. Good examples would be people from Iraq, Tunisia, and certain other parts of the Middle East. This pronunciation distinguishes between all the בגד כפת letters when dageshim are present. It also distinguishes between כ and ק , plus ט and תּ. It is consistent with use of the sheva/shewa, and pronounces ו as a w rather than a v. If you would like to hear how this pronunciation works you can see a video here. The video is done by Moshe Chabusha who is a famous Iraqi Hazzan. For a proper tutorial on the Shema using this pronunciation, with some Mizrahi melodies, you can see this video, which is also by Moshe Chabusha.
The final one is Yemenite Pronunciation. The Yemenite is similar to the Old Mizrahi pronunciation in that it distinguishes between all the בגד כפת letters and pronounces the ו as a w. What sets it apart from the Old Mizrahi pronunciation is that they have a different set of vowels, the some vowels more closely matching the Ashkenazi pronunciation, and others just being unique.
My advice to you would be to answer a few questions for yourself. Is your goal to have these kids pronounce the shema in the way that Jews originally did? If so, choose the Old Mizrahi or the Yemenite pronunciations. If your goal is to teach them a Hebrew they will use in Israel. Then teach them Israeli Hebrew, with all of its pluses and minuses. If the kids are learning Hebrew because "You're Jewish! You should be doing Jewishy type things! Just don't ask us what the point of a made up Hebrew pronunciation system is!" Then it doesn't really matter. If you want these kids to go into Ashkenazi Yeshivoth where Yiddish is the main language, then why not give them the Ashkenazi pronunciation so they stand out less when they get there? If you want them to have a proper pronunciation that still sounds similar to other forms of Hebrew, then teach them Old Mizrahi.