You can try excluding the following addresses from your proxy:
account.jagex.com : this prevents Launcher_Initialization_Failure and allows the launcher to work at all
secure.jagex.com : this is needed otherwise the launcher can't get your account details and the character selection menu will be empty
jagex.akamaized.net : this allows the launcher to complete its update check (required, or it'll get stuck and eventually error out)
secure.runescape.com : this makes the News panel work
cdn.runescape.com : this is needed for the News panel to show thumbnails
auth.jagex.com : important, otherwise pressing Play does nothing. Also required for login.
you can also try excluding *.jagex.com and *.runescape.com just in case.