There’s a song I want to identify in Justice League season 2 part 2 “Only a Dream”. You can listen to the song at 0'33 and 2'34:
I checked their OST and couldn't find it.
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Batman is humming Frère Jacques, a French lullaby. It's not obvious from the timestamps you linked (because there's background OST as well), but you can hear Doctor Destiny humming it (faintly) at 4'16 in your video.
wisea**ery sarcasm is that the song is about a monk not waking up early enough to ring the church's bells.