It would be nice if Amtrak had better options for accommodating bikes, but trains are already pretty full with passengers and luggage. Sometimes the conductor can put a bike into an outside locker of a Superliner coach, but there's no guarantee (especially on three trains).
If you don't want to rent a car, perhaps you could ship your bike via a package company? I know it is potentially blasphemous to mention it on this board, but
Greyhound PackageXpress can ship from Spokane to Winter Park. The Greyhound stop is about 2.8 miles from the Amtrak stop so it's a manageable distance on foot.
I don't know how much your bike weighs, but I did a cost estimate for a 40-pound parcel that's 5 feet long and it cost about $60 one way. The bus route is 28.75 hours long, so the bike would easily beat you there.
Downsides - You would probably need someone off the train to pick up and drop off the gear. Greyhound package would arrive in Winter Park at 12:50 p.m. while your train would arrive in Fraser at 4:30 p.m. On return, the train leaves at 10:07 a.m. while the bus leaves at 12:50 p.m. (and arrives in Spokane at 4:30 p.m. the following day).
Good luck. It sounds like quite an adventure.