Guys, thank you very much for suggestions. It took me few days to find time to try them.
First of all I got a cheap iron tip thermometer from ebay, calibrated my iron and used it at 250C.
You might try some no-joke RA flux, I use MG chems 835, pretty aggressive stuff.
I had MG chemicals 835, both older bottle (stuff is more like a paste by now) and fresh bottle. It helped in few places, however did nothing for vias affected with electrolyte. Tip cleaner worked better there. Used it very sparingly with a tip of a toothpick. It looks cool under microscope.
I bought the silver plated Kynar insulated 30awg wire and I am very impressed by it. I removed solder bridges and made all the trace repairs with this wire. My multiple attempts to get the wire to go through the hole did not work. Maybe a wire is too large for microvias. Nonetheless I was able so to solder it partially going in. Pictures are coming.
I will have to try TiNs "going through the via" suggestion another time.
In the end I had to do two trace repairs and 3 bodge wires. Luckily they all are power related, not measured signal.
It took 8 hours to repair the board and solder all ICs and capacitors back. Electrolyte compromised areas are not fun to work with.
After this I partially reassembled the meter for a quick power up test. It came up without a hitch. As I was shooting a thermal image of it, I noticed a diode CR119 getting really hot. It was shorted and somehow I missed it on initial check.
I did not have a replacement handy for this BAV103 250V switching diode. I had a 600V zener diode in a similar package (BYD57J), so I used that for a quick test. Proper part will be in next week.
With this zener unit powered up with thermals looking much more reasonable. Small U108 regulator (78L05A) was getting to 180F, but apparently it it is specked to at least 250F.
This time I did the open inputs test and got errors:
306.1, 306.2 ohms current source
308.3 2M Ohms overload
400.5 +4.48V DAC output
401.1 Regulator (not sure what this is yet).
My next action item is to check all voltage rails before tracking these individual errors.
Thank you everybody for your your help this far.