In practice, it would likely be considered forbidden for a child to help with building the Bais Hamikdash. This is due to the fact that their main 'job' is being educated, which supercedes almost everything, including building the Bais Hamikdash. And this is actally brought down explicitly in the Halacha, for example, Kitzur Shulchan Aruch 165:11:
הַמְלַמֵּד, צָרִיךְ לֵישֵׁב וּלְלַמֵּד אֶת הַתִּינוֹקוֹת כָּל הַיּוֹם
וּקְצָת מִן הַלַּיְלָה, כְּדֵי לְחַנְּכָם לִלְמֹד בַּיּוֹם
וּבַלָּיְלָה. וְלֹא יְבַטְּלוּ הַתִּינוֹקוֹת כְּלָל, חוּץ מֵעֶרֶב
שַׁבָּת וְעֶרֶב יוֹם טוֹב בְּסוֹף הַיוֹם. אֵין מְבַטְּלִין אֶת
הַתִּינוֹקוֹת אֲפִלוּ לְבִנְיַן בֵּית הַמִּקְדָּשׁ.
The teacher must teach the children the entire day and part of the
evening, in order to train them to study Torah by day and by night. He
must not interrupt the childrens' learning except on erev Shabbos and
on erev Yom Tov at the end of the day. The children are not to be
interrupted from their learning even for the purpose of building the
Now, one might be able to argue that going on a 'field trip' to see the Bais Hamikdash being built can be a great educational experience (just like we allow kids nowadays to go on field trips and do other activities that aren't strictly 'learning Torah' as part of their overall education), but my feeling is that if this exact example is brought down in Halacha as something that one should not do in place of learning Torah, it would be difficult to argue that doing so would be a beneficial educational experience for the child.
But I'll leave that debate up to those expert educators who know much more about how to properly educate children than I. May they have to deliberate on this question speedily and in our days!