This may sound crazy, but here is an idea:
Have you thought about shipping the bike via UPS or FEDEX or DHL? Maybe you could ship some parts of it ahead of time (like both wheels, handle bars, and pedals), and just bring the frame with you. It should fit in a box that would get charged like a regular bag.

Or maybe split it into two boxes, and pay for a 3rd bag. I don't bike much, so Idk how complicated it would be to take the bike apart...

If traveling domestically, FEDEX or UPS ground would surely be cheaper than $100, and you KNOW your bike will get there.

I don't have to remind anyone here, but have you seen how baggage handlers throw bags around? If you haven't, it's like watching sausage being made.