Most hypnotic musical moment in the original Jungle Book?

I don't see how anybody could make a reasonable argument for it being anything other than Trust In Me. The song sung in the first encounter is more or less the tune of "rock-a-bye-baby" if you listen carefully, and honestly I don't find that song particularly hypnotic whether or not Kaa is singing it.

"My Own Home" being described as hypnotic by the both of you is very interesting - I wonder if it might in fact have been a deliberate parallel on the part of the film.
Consider that more or less immediately prior to the climax is where we have the second encounter. It's the thing which brings Mowgli down to his lowest point, which is where he's talking to the vultures, and it helps serve as a wake-up call for Mowgli, since the one offering to let him stay in the jungle didn't have his best interests in mind, reminding him that perhaps Bagheera and Baloo actually do. After the climax of the movie and Mowgli is taken to the man-village, we have him entering his highest point, being blissfully happy and in love with Shanti, as well as being tempted by her and lured in with a song like in the second encounter, but in a 'good' way, persuading Mowgli to 'grow up' and leave the jungle, essentially the exact inverse of the second encounter.

You're of course correct regarding the power of Kaa's song, but I think that too is part of the intent - because Kaa is convincing Mowgli to do something very dangerous, trusting in him. Whilst Shanti is convincing Mowgli to do something perfectly safe, but just what initially he doesn't want to do.
The connection is just that Kaa feeds into what Mowgli wants, to stay in the Jungle, whilst Shanti is persuading him to leave it, and this is the difference in indicating that Kaa's seduction is false, while Shanti's is honest and true.
It's also mirrored in the sequel - Mowgli's trying to persuade Shanti to brave the jungle with him, whilst when Kaa finds her, he plays into what she wants like he did with Mowgli - security, protection from a friend, an authority figure to listen to. That's why to Mowgli he presents himself as an equal, calling himself his 'friend', whilst he calls Shanti 'little one', belittling her.
I.e don't trust people who just tell you what you want to hear.

What I mean is that Shanti genuinely loves Mowgli, whereas Kaa's seduction is so he can eat Mowgli. Plus being made when it was, and considering the original book's message, it's clear the film at least wishes to portray Shanti 'civilising' Mowgli as a positive.
Regarding Shanti manipulating Mowgli, another interesting parallel between her and Kaa is that a note is made at the beginning of Jungle Book 2 of Shanti's "beautiful brown eyes". Plus the teasing of Mowgli for falling under Shanti's 'spell' by their adoptive families.
As far as Mowgli being literally under a spell, I think the original movie shows him looking back and reconsidering before making his decision to enter the man-village, so he's not entirely coerced. And in the second movie, there's plenty of times where Mowgli is bullying Shanti, I'm sure if she could put him under a spell then she most certainly would do so again to get him to stop.