The halacha is as you said- one cannot command a katan to do a melacha for an adult, but one is not mechuyav to be be mafrish a katan below the age of chinuch from doing a melacha.
You claim that a 1-year-old has the daas to understand a father’s ratzon is ee efshar leholmo unless your son is an ilui atzum. The halacha is a child can sleep in his parents’ room until 18 months old. Clearly, a child until that age has no da’as whatsoever. A 4-year-old is below the age of chinuch, but would understand what his parents want. My toddler has no clue what I want if I don’t tell him directly. Rabbi Simcha Buni Cohen brings down this halacha on pages 66-67 of his sefer “Children in Halacha” and writes it as a practical halacha, keneged your Daas Torah. He also brings down that, in cases of great need (but not stam), one may instruct a child to do something if it is for the child’s benefit. Not that I would rely on this unless I asked my rav, but it’s not pashhut to call it a melacha deoraysah in the case of a shul where the child benefits from it as well.
Here is Rabbi Zev Leff on the same point:
http://www.rabbileff.net/shiurim/answers/1000-1249/1172.mp3
I will also add that in my house, all bulbs are fluorescent and not incandescent.
How old was the child in the shul? Was the power just for fluorescent lights or also for incandescent bulbs? Was it pashut that his child understood what it was for? Did the child benefit from the lights being on?