One of those reposts where I'm just glad to see it again because it's so good lol.

[I'm OP, just under a different account.]

I forgot to mention a small detail: the *bʰāt- root for "boy" seems to be an old reconstruction. Usually Proto-Indo-European roots have three consonants, and even short *a is surprisingly rare - and it's mostly from borrowings, like the Semiticism *táwros. There's a Proto-Slavic root that might be related, too.

Another possibility is baby talk, given the original meaning associated with "father". That would explain the weird vowel and structure.