A sous-ventrière is a cinch that passes under the belly of a horse. The expression means more or less eating and drinking so much that the belly (and any cinch around it) is ready to explode. It is a familiar expression that apparently was quite commonly used, but I have only read it in books, younger people are not that used to horses – and I’m not that young myself…

Some now more common expressions:

• je m’en suis mis plein la lampe
• j’ai les dents du fond qui baignent (may be a little more vulgar)

But à m’en faire pêter la sous-ventrière is still understood by native french speakers.

Indeed, this expression is an equivalent of “I’m stuffed” in a rather colloquial way, but frankly, I’ve never heard anyone saying it.

Most native French speakers would understand, but they’d believe you made the expression up (which you can definitely do).

In your sentence, “J’ai bien cru que” definitely asks for an image, and the structure itself makes it colloquial. What I hear the most is “J’ai bien cru que j’allais exploser” (literally, “blow up”).