I think @mathpro did a tour last year during the CNY holiday. Maybe he can chime in. His account was posted here: Taiwan round island cycling (11D)
Hotel situation on the west coast would be an epic nightmare, somehow mathpro found a place AND was cheap. Based on stories I’ve heard from colleagues, I would still plan ahead and see if you can book in advance. If you have friends, stay with them.
As for the east coast, some of the small bnb’s might be closed during the holiday, it doesn’t hurt to plan a route and see where you might end up and check if bnb’s will be open in that town. Most of the time, you don’t need to make reservations for bnb’s on the east coast. I would say CNY holiday is a special case.
Traffic should be ok, as long as you stay out of roads that take cars on and off the highway.
My suggestion would be, if you have no obligations during the first few days of CNY, go on day 3. Most people will travel on day 2 to avoid the day 1 crowd. This is my observation through friends. I’ve never made the trip south since my entire family is in Taipei.