After doing a little research, I found a fairly easy method to disable the annoying “Message Sent” notification on windows mobile devices.
First, download / install a registry editor for your mobile device. I had good luck with (free) PHM Registry Editor.
Second, navigate your newly installed registry editor to:
HKEY_LOCAL_MACHINE\Software\Microsoft\Inbox\Settings\
If the “Settings” branch doesn’t exist, create a New Key named Settings. In your newly created key, create a new String Value named SMSNoSentMsg, giving it a value of 1.
Perform a quick soft reset, and you should be good to go!