I registered just to reply to this thread. I've been so frustrated with this as well, and I just found a fix over at a different forum (not sure if linking is allowed so I won't be posting the link to be safe).
Basically, you can download a file explorer app (e.g. ES File Explorer), then when you choose your ringtone inside the Twitter app choose that app instead of the default 'Media Storage' app.
If you already have a file explorer app installed, you will have to clear the defaults of your 'Media Storage' app first via Settings -> Applications, and then Twitter will ask you again which app you want to use for choosing the ringtone.
As for the ringtone itself, I wasn't able to directly access all the default ringtones because it only seems to have access to the sdcard folder, but it should be possible to go into the actual file explorer app, copy the ringtone you want from system -> media -> audio -> notifications, and paste it somewhere inside sdcard folder. Then you can go to Twitter -> Ringtone -> select file explorer, and choose the ringtone you pasted. Voila!
Sorry if my thoughts seem a bit disorganized. Just so hyped that the issue was finally fixed after months of cringing every time my Twitter notifications start to get spammed.