Im dealing with the same issue, but my public ip is from 185.xxx.xxx.xxx range. And when i change my network to mobile hotspot (with public ip 188.xxx.xxx.xx) i can log in to my multiplayer server without any problems, and launcher remembers my login and password.
I found out that this issue is related to the public ip adress. When i connect to internet via my mobile hotspot and my public ip adress changes, I can normally join servers and there is no problem with refreshing access token.
I have the same problem as @Liam Owen Wolfe.
Launcher can't refresh my access token (Error message:Failed to refresh login. Please login again.).
My launcher_log.txt looks exactly like Liam's(Session: Failed to refresh token).
I have tried the same methods as Liam - updated all drivers, windows 10, java etc., checked if i can log in from another pc, even changed password, but with no results.
The bug is related to public ip of your home network. If you have possibility, change your public IP adress (in some cases simple router restart works). See MC-130115 for more informations 🙂