"No battery present" - HP Elitebook 8460p x64 Win10

Hello!

I have a puzzle, sorry in advance.

I have an HP Elitebook 8460p (x64) laptop that I bought refurbished in March 2020, with Windows 10 pre-installed.

The battery has always been weak, but for the past few weeks, Windows no longer detects any battery at all (even though it's inside):

I ran a "Battery Report" from Windows Powershell, I ran the troubleshooting user, I did tests in the diagnostic tool before the BIOS, same result: no battery detected

Yet it's still inside, and now the PC only works when plugged in.

I assumed the battery was dead, so I bought a new one: GREENCELL CC06 Notebook Battery 6360b

It was clearly stated as compatible with my hardware.

New problem: when I put in this new battery, the PC doesn't turn on at all, even when plugged in.

So back to square one, it only works with the old battery that Windows doesn't detect, and only when plugged in;

Then I assumed the issue was either the new battery or the PC.

I tried to update the BIOS since it seems old ("Hewlett-Packard 68SCF Ver. F.60, 12/03/2015"; SMBIOS Version - "2.6") and apparently that could solve the problem.

New problem: the downloads on the HP site are not available for Windows 10 on this PC ->

No worries, I took the latest one for Windows 8.1 x64 hoping for the best:

But when I went through the procedure to update the BIOS, here's the error message I got at the last step (= at the time of reboot):

So the BIOS update seems impossible due to my power issue, while I was trying to resolve my power issue by updating my BIOS.

Can someone help me with this?

How can I find out if it's my computer or the new battery that's the problem?

I also want to mention that I tried the "purge" process (turn off, remove external devices, remove the battery, press the power button for 40 seconds, put the battery back, plug it in, turn it on), without success.

Thank you very much! I'm available if you need more information.