Kuo Kuang is usually pretty cool about letting you put non-bagged bikes in the luggage compartment with the wheel off. Not all buses have big luggage compartments.
However, from Yilan, you will be taking Kamalan Bus company through Snow Mountain. They require the bike to be disassembled and in a bag. This is a new operation with high standards. They mean it. Also, tickets sell out real quick on Sundays, so this is not always a reliable option.
From Yilan, you could also bag your bike and get two of them along with two passengers into the new mini-station wagon taxis. That would be NT$1200.
Ranging further afield, Ubus (Tonglian), for some time now, has only allowed bagged collapsible bikes. Absolutely no full-sized biked, bagged or not.
The whimsically named Ding Dong (Taidong) and Hualien Keyun will allow bike with wheel off in compartment below but will charge extra half fare.
Jiayi and Nantou bus companies have been way cool about allowing bikes in various configurations on board with no extra charge.
But the very cool news is that there are now a few local trains that you can take your bike on without bagging it (roll on, roll off). On Sundays, coming up from Yilan, these are:
區間車 2732 – 蘇澳 至 樹林 16:33 18:28 1小時55分 [折疊自行車可置於攜車袋中隨身攜帶] [自行車運送服務]
區間車 2744 – 蘇澳 至 樹林 18:54 21:01 2小時07分 [折疊自行車可置於攜車袋中隨身攜帶] [自行車運送服務]
The departure times are from Yilan Station.
There are some restrictions–you can’t use every station (Taipei station is out) and there are some limitations on availability but basically you just show up and it will work.
Here’s the new TBA Bike Train page (Chinese only) where you can look up bike trains in other regions.