So, an update for those curious:
Sony told me to send the camera and the battery to Precision Camera. After a few weeks in "Engineering Delay" I finally get an email from them with an invoice for $1,119.33 that says this:
"Technician Findings:
Suggest replacement Battery
Chamber, Battery Lid, Front Grip,
Grip adhesives
Battery leaked in chamber, Grips are
peeling"
Notice the "battery leaked in chamber" part.
A few hours later, both Sony and Precision Camera claim that the battery DID NOT leak, and the damage must have come from outside.