The connector and cabling from the battery to the motherboard doesn't get the stress like the AC adapter connector does, but it is still possible for it to fail, although that is unlikely, especially with multiple batteries. It should be easy to test though. If they can charge the battery outside of the laptop and install it, it should show charged in the software if it doesn't, they would need to fix that first.

There should be a decent repair shop in Silicon Valley, but I don't have any information on one. You would need to do some checking. There is no guarantee that the factory authorized shops would do anything other than parts swapping either, although if you do send it off, make sure they know you want info on the issue communicated back and will pay only for a fully repaired laptop. Back up your data first of course.