The point of Galfert paragraph that I quoted it that to service to outdoor console without sending erroneous data to the sites you send data to, ie Weather Underground you can remove the batteries from the console but with the WS- 2902A this would lose the hard earned barometric pressure settings.
If you turn off your router these signals will not be sent out to the sites that you send data to so you do not need to remove the batteries from the WS-2902A and you will not lose the hard earned barometric pressure settings.
correct - but you still need to correct the erroneous data somehow (unless that's unimportant for you).
They won't be sent out, and they will not be sent out later - only what's the situation at the current time will be sent out once the server connection is reestablished.
What galfert meant (how I read it/him) is that in your console memory you still have the wrong data of that servicing period - and either have to correct them or leave them as they are.
The rainfall total data, as galfert pointed out, will need to be corrected anyway.
Not sure if you can correct the other wrong data, maybe you can delete them. In the WS2000/5000 console that would be the case. But it would be the whole record(s) for that period. The WS-2902 I don't know well enough.
Taking off the batteries from the console would mean having no wrong data (having no data for that period at all), but, as you say, at the cost of losing the barometric forecast for some days (and possibly also the calibration settings).
For you to decide which path you want to go down.
As we say in my language: you cannot get around having to die one (type of) death
The only other (rather hypothetical) way how to do would be taking off the batteries from all your sensors and not from the console - then there are no data recorded, or only empty data (--) recorded.
However, with a WH65 as outdoor array this becomes very difficult as taking off the batteries won't change anything. The high-capacity capacitor, once fully charged, will usually take more than two days in a dark room for itself to run out of power. Maybe after a few very short, cold and overcast winter days it will be expired and run on battery power only. But difficult to find out with layman's means when this is the case.