I noticed that the power connections matched that of a common LM358, so with little to lose, I just tried one. It sortof works - the reading increases when the input increases but it is uselessly nonlinear. (That chip is near the output of the module, so I guess it takes the square root or something.) Looks like time to replace that module with a LTC1968 circuit since the rest of the meter is in good condition. (The previous owner even replaced the batteries! I replaced the constant current charger with a smart charger so it doesn't overcharge the batteries if left plugged in for a long time.)
Some signal tracing with a scope also found a bad 4016 in the AC signal path, but the preamps were fine. My guess is that the previous owner must have damaged the 4016 and 473777 while replacing the batteries (most likely ESD or an accidental short circuit), then threw the meter away in frustration after it fails to turn on with new batteries. The meter works great on all DC and resistance measurements after I removed the shorted 473777.