TL;DR:
C162 had gone bad as well. Removed it.
Does anyone know the capacitance of C162?
The ribbon cable connector leg for the V+ trace was destroyed. I bypassed it by connecting directly to battery terminal.
Used through-hole instead of SMD to replace R16. Soldered directly to IPL chip.
Need to attach C162 replacement between resistor and ground.
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It's been quite a journey, but I think I have it fixed. Thanks in large part to
this post by dilav and help from my brother-in-law.
First, we realized that the left-most leg of the ribbon cable was also destroyed by the corrosion. This connects V+ from the battery on the controller board. Failing to "fix" the leg by flooding the gap with solder, I decided to just connect directly to the battery terminal.
Then, being unable to solder the tiny 0402 footprint, I instead used a 120ohm through-hole resistor to replace R16... At least, I would have except I did not have any on hand so I used 6 of the 100ohm I did have. It's temporary I promise!
I connected one end to edge of C162 and the other to the V+ battery terminal on the controller board. This worked! For a few hours... The next day the calendar was broken again. The battery measured 0.22 volts...
V+ was shorted to ground causing the battery to drain quickly. I started mapping out traces looking for short.
I found that C162 measured only 5ohms across. Clearly that cap was also killed by the corrosion and causing the short. I removed C162 and this time attached the resistor directly to the IPL chip (this was much easier than soldering to edge of SMD).
Success! No more short to ground and calendar is still working days later.
I still need to replace C162. Should be easy to attach a through-hole cap between resistor and ground. Does anyone know the capacitance?