Find My app won't add Apple Pencil (2nd Gen)

I lost my original Apple Pencil (2nd Gen) and finally got a new one through Verizon. So, I wanted to make sure I added it to Find My this time in hopes of avoiding losing this one.

However, Find My simply won't acknowledge it's existence. It has my iPad and that's all. And there isn't a "Add Apple Pencil" option when I press the + in the device list. But I have tried the "Add Other Item" option and it simply searches endlessly.

Yes, my Apple Pencil is paired and working correctly.

Yes, my Apple Pencil is fully charged.

Yes, my Bluetooth is On.

Yes, Find My has location permissions and my Pencil is attached when I try to add it to Find My.

My iPad OS is updated to 18.0.1 and I've already tried all the suggestions I've found anywhere:

Attach, restart iPad, unpair & repair.

Unpair, restart iPad, repair.

Uninstalled and reinstalled Find My.

Reset iPad settings and reset-up Apple Pencil to my iPad.

And about a million other minor adjustments or ideas I've found online.

Absolutely nothing has worked. Please help.