I've worked on 1062s for many years. It seems that acid flux was used to solder in the LM309K (on them all) and it was not cleaned off.
On your other (working) unit, check the 10uF/350V (if OEM) capacitor on the back panel. I've replaced that cap in maybe 90% of the units I've worked on. The display can look fine even if this cap is shot.
Also on your other unit, check the six blue (if OEM) electrolytic capacitors up front on the DCV board. I've replaced some or all of them in possibly every unit I've ever tested them in.
For the battery, I've replaced a few, and there is a recommendation to replace them every five years, but I don't remember actually testing a bad one (in the 1062), even if it looks OEM. It seems that the (CMOS) circuit draws so little current (Nano Amps), that it's the shelf life of the battery that matters most.
If you try to power the battery circuit, I'd use a current limiting (and isolation) resistor. The circuit draws Nano-Amps. I personally would not worry about it. I sometimes clear and reprogram the constants anyway.