Repair advice for Starrett electronic indicator, failure to power on, freezing

[jp430bb]

I've got a Starrett 2730-0 electronic indicator that I picked up used on eBay.  With a 1-inch range and 0.00005-inch resolution, it's a sweetheart...when it works.  Sometimes, I have to hit the power on button five or six times before it will power on.  Then it'll work fine for 15 or so minutes, and then it'll freeze. 

I disassembled it, and along with impressively small passives on the button & display PCB, I could see that some previous owner had used alkaline LR44 button cells in it and let one spew its chemistry on the inside.  Some of the PCB tracks are visibly corroded under the solder mask. 

After cleaning the PCB with contact cleaner, it seemed to behave better, but only for a month or two.  Now, it's back to being flaky.  Any ideas on what I might try to repair this thing?  I haven't started with checking voltages or anything yet, but with such small parts, it's looking like quite a challenge. 

By the way, this unit is the kind of thing that I think of when I see test gear on eBay "Tested to power on only"!