Hi Rami,
thanks for the hint.
I have a bit different situation:
In the above case there was not internet connection at all (Play store timeout, Chrome not connecting).
In my case I have internet connection (Play store working, installed ESET MS via Play store, Chrome working, can connect to www.eset.com), but ESET MS can't activate and doesn't send "Eset support info" logs to Eset server.
Anyway I have tried to connect to unsecured wifi (I have nothing to loose) and the situation is the same:
Play store online, Chrome ok, ESET MS not activating, ESET support info not sent.
I will investigate the Android rooting and upgrading options.
BTW I have downloaded some (a bit outdated) Android source code for the Panasonic FZ-A1 directly from Panasonic public ftp, so maybe some more skilled in Android development can check options for upgrading to newer Android or building LineageOS build.