I have gotten as far as to finding the location of the ringtones.
You’ll have to SFTP into your phone, then navigate to the /System/Library/Audio/UISounds folder. Here the SMS ringtones are labeled “sms-receivedx.caf” where “x” is the number of the sound.
1 is Tri-Tone, 2 is Chime, 3 is Glass, 4 is Horn, 5 is Bell, and 6 is Electronic.
It doesn’t seem that the name is embedded into the file, so I am not sure how it sets that up, probably through a special file somewhere.
Hope that helps.