Ask LLOG: "I's" ?

Email from JK:

My daughter remarked that on several occasions recently she has heard presumably college-educated people use the word “I’s”, in a sentence like "I just picked up Margaret and I’s cat from the vet."

I'm guessing the reason for this jarring (to me) usage is that the sentence could be more logically parsed as

I just picked up Margaret and (my cat) from the vet than as
I just picked up (Margaret and my) cat from the vet.

The brain turns to I just picked up Margaret and I’s cat from the vet as a solution because because Margaret and I is a strong connection like my cat, and it's encountered first in the sentence.

So it's parsed as I just picked up (Margaret and I)’s cat from the vet.

That sounds awful, but is it acceptable usage (obligatory SMBC reference)?

There's quite a bit of discussion about this on the web, e.g.

"Pronouns and joint possession"
"My Friend and I’s Grammar Problem: Joint Possession with Pronouns"
"and I's or and my?"
"Possessives and attributives"

The earliest example I've found is in the subtitles for an episode of “Married with Children” from 1992:

We're busy over here now, Bud. It's your mother and I's quality time.

[Found via the COCA corpus, which has 67 other examples]

But I wouldn't be surprised to find examples from much earlier dates.

And we shouldn't forget the other source of I's, as in this phrase from the 1921 play Granny Boling:

I's lived a long time, boy, and I's seed a heap of trouble.