Taiwan has some big mountains; more than what most people would think. They do get some snow, and thus some snowmelt, but for the mountains around Taipei and that time of year, you need rain to make the rivers and creeks run. Taipei has granitic mountains on both side of it (you will cross one range between Taoyuan International and Taipei proper) so if you can catch the day with rain you might be able to find some things. Might have to get creative but I bet you could swing something? Given your extremely limited timetable and a necessity for rain I don't think you can afford to be picky and only find IV/V. If the only option you can find is class III, I would say do it.
The river that flows past the National Palace Museum (I forget the name) looks like it could have some ok rapids (and maybe a dam or two?) and I think you could use the city buses for a shuttle. Those mountains (the Yangminshan) also look like they could have several other decent creeks, there are some longer drainages on the north slope that looked possibly feasible; roads along the upper parts of the drainages and at the bottom, and not far from the rivers should an epic happen. The good thing is that time of year can be fairly rainy, the down side is thats smack in the middle of typhoon season. Not sure if you've been watching the news but if not, google "Soudelor", they just have a really strong typhoon season roll through and fuck a bunch of shit up and kill some people. But without rain the rivers will be bone zone.
Seoul is surrounded by many, many granite hills, and in some places rafting is quite popular there. Google "south korea rafting" and you can find lots of beta. However, just like Taiwan, you need rain to make any of the low-elevation hills run. September is just coming off their rainy summer season as it slides into fall, so you really could have any kind of weather. There is much more of a whitewater infrastructure in Korea so I would imagine it may be a touch easier to find gear there?
Good luck and let us know how it goes, if you make down anything and anything else. We love TR's! If you didn't already know, do be aware both airports are a solid hours drive by taxi/bus to the cities themselves. Either way enjoy your trip, both cities are wicked cool and worth exploring around. In Taipei the night market between Shilin and Jiantan MRT stations is pretty cool and has some delicious food!
"Don't f$&@ing eddy out, just run it! Whaddya doin??" -LMyers