In fact, I believe they now do both. Starting in 2005, Taiwan has started using the name "Republic of China (Taiwan)."
Not that it really matters, but I wonder which would actually bothers the PRC more. I would have thought that the real source of irritation with, "Republic of China," is that it references the name used by mainland China up until the revolution of 1949, thus manifesting Taiwan's historical claim to be the legitimate Chinese government in exile.
CCP wants Taiwan to claim to be China as then it is seen as an internal struggle between two Chinese factions. So the CCP takeover of the island becomes reunification. If Taiwan calls itself Taiwan and develops a separate identity at some point it becomes a struggle between two separate identities and it would not be a unification but a hostile takeover.
To sum it up, most important for CCP is that Taiwan is considered Chinese. I know CCP likes to play the long game, but at some point Taiwanese cultural identity will have shifted too far for this to work. I think it is close that that point, and that in another generation it will most likely be there.
Not that it really matters, but I wonder which would actually bothers the PRC more. I would have thought that the real source of irritation with, "Republic of China," is that it references the name used by mainland China up until the revolution of 1949, thus manifesting Taiwan's historical claim to be the legitimate Chinese government in exile.