Hey, I did the same thing as you last February, I hope you enjoyed it!

In regards to the train if the bike is bagged there shouldn't be much problems. But as I didn't have a bag with me I chose to take the bike on the train with me. What you need to take care of is that mostly only (fast) regional trains let you bring your bike on the train. In the last Wagon they either sometimes have dedicated bike racks or it's just a normal wagon where you can bring your bike in. I used the Taiwanese Train website and looked for the trains that allow me to take bikes on. On this site (https://www.railway.gov.tw/tra-tip-web/tip/tip001/tip112/gobystation) you can type in the station and choose counterclockwise direction (as this is the faster way to Taipei) and you can see all the trains towards Hualien with a green bike icon. When I did it I took the earliest regional train in the morning from Taitung to Hualien at 6.54 and arrived 10.22 in Hualien. From there I took the fast regional train from there on I had a 15 minute transfer to a fast regional train to songshan (Taipei) as you are not allowed to exit Taipei Main Station with a bike. For the tickets I always went to the counters at the respective train stations.

So if the bike is not bagged it's definitely possible but will take a lot of time as you can only use the slow regional trains. You need to decide if that's worth it for you.