Hi there.
I've not heard of it being allowed in the overhead. It's not a small seat.
If you bought a ticket for your baby, an angle adjuster, and got a seatbelt extender from the flight attendants when you get on board, you could use it rear facing. Then your baby would have a seatbelt, like you do. You're right to be concerned about checking the carseat. That's not a safe option. Baggage handlers aren't known for being gentle with baggage. But, I'd be a little more concerned about your baby. The plane won't push back unless my seatbelt is done up and my laptop is off my lap. For my protection and the protection of everyone around me. There's nothing that protects your baby or everyone around you with your baby on your lap.
Have you seen the FAA's site on the matter?
https://www.faa.gov/travelers/fly_children/
Definitely practice installing the Rainier before you travel. It has a bit of a learning curve, and can be large rear facing. Make sure the rear facing base and angle adjuster come with you.
Wendy