Even after adding Send As permissions to an account, Exchange will deny your ability to send as that address. In order to force Outlook to download the latest Offline Address Book which contains permissions for all of the addresses in Exchange, you need to first close Outlook, then navigate to C:\users\appdata\local\microsoft\outlook\offline address books\
and then move the folder that should be a jumble of numbers and letters out of that directory structure. The next time Outlook starts, it will be forced to download a new instance of the address book which will then have the correct permissions data.