It sounds to me like cutting the shaft is the answer (YT video on page 6 previous, I think).

I'd be using an angle grinder, rather than the super $10 reciprocating saw blade. Even though it sounds impressive. Now, it'd kill me that I was ruining my chances of getting my core charge refunded, but the AutoZone listing that you shared (page 1) didn't indicate that there even was a core charge.

Once you cut the shaft, the carrier bearing no longer poses a problem. The remaining part of the shaft ought to come right out of the transaxle (knock on wood). You will need the carrier bearing when you reassemble things, but with that part of the shaft sitting on your workbench, all it'll require would be some vigorous tapping with a BFH.

The YouTube comments are universally positive that this is the way to do it.

And needless to say, you're obliged to report to us when it's all put back together.