Tucson -> Zurich apparently means 2 connections each way.

1. Bike fees often apply to each segment. I'd suggest that you call your carrier to validate the actual fee that will be charged.

2. Flying bikes on routes with several connections increases the likelihood of baggage delays, especially when layovers are short.

3. You may want to consider buying/renting in Zurich. Would make your trip much easier. I, like you, much prefer to ride out of the airport, but in your case, it involves significant cost and complexity.

See here for a few options.