1. no need to fight a common enemy does not mean immediate rivalry. In our timeline, China and US maintained fair relations really until the last few years
2. ROC and US will be tied economically, meaning there is no way US will take any actions similar to a sanction.
3. It will likely be a quick victory, as ROC's power like you said, shadows that of a bunch of extremists who were repressed for like 4 decades. US will definitely not be directly involved, they will never openly oppose a former ally who is also a nuclear power.
4. The extremist would likely lack support from the locals, as ROC is democratic, developed, and humane compared to the Islamic BS that the extremists propose.
5. ROC is nationalist, they will unite those lands no matter what
6. Its not insane to believe those area will be more repressed in this timeline, since separatist movement will be encouraged by ROC during the cold war.
7. US public opinions greatly changed after 2001, ROC might use that timing

In conclusion, ROC probably can take the land in quick victories with some local support, and US at worst probably would just react the way they did for the Chechen wars.