"Basically what I'm trying to do is to get all of the shared text styles, then assign all of the new values(font family/weight, etc.) from the theme(plugin) and send it back to sketch.
The problem is that some of the layers/symbols get updated automatically but others don't. Some of the layers/symbols that point to all those text stylings note the change without updating it directly so the user needs to go and apply them by themselves by refreshing the layer/symbol manually."
This to me is the meat of what you are trying to accomplish, and it is totally achievable. I think the talk about the "sketch library" is what is confusing the matter, because at the end of the day, you said "a custom sketch library that the user imports" which is no different than taking operations on an opened sketch file.
All that said, what you are experiencing is correct and what I have experienced as well. It sounds like you have successfully modified the shared style, whether it be in a shared library, or the same file, but are noticing that layers in which that style is applied are showing as being "out of date" requiring the user to refresh. This is because you also need to apply the same modifications you made to the shared style, to the object in which the style is applied. So if you change the font of the shared style, you need to change the font of the style of any text layer for which that style is applied, so they remain "in sync". This means you need to iterate through all objects that have the shared style applied, and make the same style modification. This can be done via plugin or externally directly on the file contents.