How I would imagine this should work would be to “Edit Existing Character” and then “Edit Export File” of T1 (from the example we were talking about earlier). Do what you need to do in here and then re-publish the character as T2 like you mentioned before. I wouldn’t save the file directly but rather republish the character by clicking “Build Control Rig” at the end of the “wizard” and either saving it with a new name (which is basically save as) or overwriting the existing rig I’m editing (which is basically save). Saving this file as is (without renaming it) should do the same thing as republishing unless you have added joints or something so I don’t really think there is much difference when it comes to that. But if you want a second character (T2 in this case) I think you need to republish so the rig file gets built. If I remember right the rig file is what you use when you’re animating and if one isn’t created then you won’t see it in the menu for animation.
I hope this helps.