If you’re leaving from Taipei to head down and start in Hualien, the Giant store on Nanjing East Road near Nanjing Fuxing MRT rents touring bikes.
You can rent from them and have them pack it up and sent down to the Hualien Giant store.
OR you can rent directly at Hualien. Some people have done the above process because the Hualien store didn’t have any bikes available at the time.
Now the tricky part is returning the bikes. You can rent in Hualien and return the bikes in Taidong (and vice versa), but I haven’t heard of anyone returning in Kaohsiung. You would have to give these shops a call/message and see if you can return the bikes to a Giant shop in Kaohsiung.
If there’s a language issue, I remember the Taipei store on Nanjing and the Hualien store both have English speakers.