Replacing the battery would appear to be the correct decision since the screen problem doesn't happen with the charger connected.
You say the WiFi drivers (I assume) were recently updated, were the updated drivers Asus Wifi drivers, Windows generic WiFi drivers or the WiFi card manufacturer's drivers?
It is always advisable to have the Asus drivers installed on Asus equipment as they know their equipment best and may have adapted the drivers to suit their particular requirements.
If Win 10 is the installed OS here is a link to the latest Asus WLAN drivers. Select the drivers appropriate to the make of card that is installed, see Notes near the top of the webpage in the link regarding installing the drivers.
Try installing a free WiFi scanner program (there are others online if this doesn't suit you) just to see what is going on with your WiFi connection regarding signal strength, number of networks using the same channel which may be causing interference etc.
Don't know about reflowing the cards, seems a bit extreme considering that you haven't proved where the problem is yet. It may be unrelated to the actual WiFi card. Initially it may be more prudent to invest (or even better if you can 'borrow' one from a friend) in a cheap USB WiFi adapter and see if the connection suffers from the same problem or not to give you an idea of where to go on from.
Just some thoughts.