There's a good chance this is an easy repair! Do you have access to a soldering iron and another multimeter?
If you look on page 139 the manual PDF (
https://assets.fluke.com/manuals/8060a___imeng0300.pdf) you'll see the schematic for the main board of the meter. The part we're interested in, the power switch and such, is tucked away in the bottom-left corner. VR3 is a 12 V Zener meant to protect the meter from reverse voltage such as accidentally reversing the battery. In this case VR3 probably failed short, which means it did its job perfectly!
Turn on the 8060's power switch, and then with the other multimeter measure the resistance across the battery wires in
both directions. If VR3 failed short, you should see a very low resistance in both directions. Page 135 of the PDF shows where VR3 is on the board: placed vertically and tucked under the display, right next to the battery eliminator jack. If you cut or unsolder the wire coming out of VR3 and the short on the battery input goes away, then you've found the cause. Remove the original VR3 from the board and
replace it with a new 12 V Zener diode to restore the 8060's reverse polarity protection. The meter will work without VR3, but if you accidentally reverse the battery again and VR3 is missing you
will cause irreparable damage to the meter.