Here is the way I did the thing, of course first time I saw nothing on the display and decided to swap the testing leads in between them... it was too late anyway as first time I connected the GND of the battery to COM lead...
As I've read here on the forum, the COM lead is very low impedance (like DSO GND) and you've gotta be really careful where you place it. Probing just with the V+ lead would be harmless (impossible to measure current without COM of course, as my intention was...)
I have a hot air soldering gun, flux, solder wick... I've soldered even smaller packages, so no issues here.
The only idea would be not to damage even more things while replacing the ADC and ASIC. Even if the ADC seems fine, it's worth ordering that thing too, just in case.
But, I have a stupid question here, the ASIC comes already programmed, doesn't it? I know these things can do only one task... from Bitcoin mining
p.s. The idea of the whole topic is to try to fix those things and help other people... I could just go and buy a new meter and that's it, learn my expensive lesson, but that won't make the forum interesting, wouldn't it?