I skipped a bunch of this... Yes that is a battery, and probably a rechargeable battery. In many cameras if that battery goes flat, the camera fails. And once that battery goes flat, they do not recharge again. Our JVC cameras do this (though the camera stays working) and GH2 cameras seem to completely fail when it goes dead. They are super cheap, but can't be certain the camera will work after it is replaced.
I would contact Dave at the EEV Blog and see if he would be at all willing to take a look at it and make a video about the repair, he might give you a decent price if he thinks he can fix it.
https://www.eevblog.com/ under "ABOUT" is a contact page with some emails, you can also find him on Youtube. Good thing is that he is closer to you than some other paces.
After that, it is probably a brick. One thing we've been having with our JVC cameras is condensation under the BGA chips, that's a $900usd service to replace the main board. Or at least they say it is condensation, typically under the JVC branded encoder chip.
The lack of service manuals will stop me from ever buying BMD cameras for work. If they don't respect me enough as a tech., then I won't put my stamp on the cameras. Never had an issue getting or buying service manuals from JVC, Sony, or Panasonic. But these are always cameras in their professional or broadcast product lines. And we are looking to buy 12 new cameras in the next couple of years.