Cycled this route several years ago.
Never stayed at an official camp site, just made use of pavilions and rotundas and the like.
Sun Moon Lake - bird viewing platform near north end (or end near where water gushes in)
Toward HohuanShan coming out of Puli - roof of a church in a small hamlet just off the road. Father (?) of church gave us permission but only after 20 mins of him trying to persuade us to stay in his house, eat his food, use his shower, etc. Lovely old guy.
Tayuling - just before the road forks off to Lishan and Taroko, there’s a long flight of steps going up the mountain. This path leads, I think, to one of HoHuanShan’s peaks but just at the top of the steps is a lovely pavilion with awesome views down the valley and across to the scary mountains over the way.
Taroko Gorge - slept down in the Wen Shan hot springs. They were closed to the public at the time.
All free, no one ever bothered us, quiet but amazingly beautiful. Filled water bottles from road side waterfalls, ate cheap and simple food from road side vendors, built campfires and supped tea as the sun went down.