Resolved this issue and just wanted to follow up on how it played out.
As I previously said, AAMCO got the part out, but they said it was very stubborn, and it pulled out the sleeve that slips into the transfer case. They got a remanufactured axle and installed that unit. They didn't say anything about the sleeve, and it never really came up in our discussion. The guy I talked to in the front office didn't do the job so he didnt know anything about it, but it seems though that it wasnt any real issue re-installing it. They slathered everything up in anti-seize and installed the new axle. The car rides smooth again so I'm glad that fixed it, though it cost me $600 (~300 labor, 300 part). It sucks that I had to pony up so much money, but I could not do the passenger side alone, so it was worth it. I attached a picture for all to see. As seen in the picture, the security bolt secures the sleeve and the bearing. The c-clip secures the carrier into the sleeve, and the security bolt secures the sleeve/bearing into the transfer case.
I think the previous comments may be right, in that the sleeve was not meant to come out of the transfer case. But in my experience, it doesn't seem to be a problem to reinstall it, as long as nothing is cracked or damaged. Just for reference, here is a video of a fella that had the same issue as me with the 2004 Toyota Matrix, except he cracked the transfer case when he tried pulling the axle/carrier bearing/sleeve out. https://www.youtube.com/watch?v=BUuEVCypT1s
So if anybody tries to service this and has the same issue, I would use penetrating oil and giving it a lot of time to soak in. AAMCO used heat and then an air hammer to jackhammer it out. When reinstalling, be sure to use anti-seize! I'm just glad to be done with the front CV axles, but I have to turn my attention to some other issues now.