Congrats on being an original owner and keeping this game alive for 40+ years. As soon as I started reading I mumbled to myself, original battery!
Since your game is still working it might be worth an attempt to restore the original board(s) rather than replace with an Alltech or B&I board. To service the original MPU you will need to immediately remove the battery and mitigate the damage by cleaning with a mix of baking soda and vinegar. (lots of other home brew cleaning solutions out there besides mine). Order an NVRAM chip to replace U8 on the MPU eliminating the need for batteries forever.
If your board does not survive surgery both Alltech and B&I (Weebly) offer quality replacements.

The other issue is the power supply connectors are burned out. That is normal for Bally games of this vintage. I have been ordering kits from Big Daddy which include keys. Getting a connector key wrong or leaving out could be death to your game. You will need a proper crimping tool and should practice with it before beginning the repining. Do this one wire at a time. Then compare to the schematic wiring guide.

Good luck. -Al-