There is a new version of WebLaunch available which implements a
new way of getting notified when the power button is pressed. It no longer uses the goingToScreensaver notification and dbus-monitor so the Kindle shouldn't go to sleep anymore.
It would be very nice if somebody with a PaperWhite could test it as I don't have one and it is mainly developed for it. If it doesn't work it would be good to check if /dev/yoshibutton exists.