Where and how the bunny rides is up to the airline. However, bunny will have to go through security.
I did a 6-month gig as a trainer for TSA screeners so I can provide some insight on the security issues…
If they are doing it right, they will ask you to remove the bunny from the carrier and send the carrier through the X-ray machine. You will then walk through the metal detector holding the bunny. If there is an alarm or if the screener feels it is needful, they may subject both of you to "additional screening" which can mean a hand-wand or a pat-down at the discretion of the screener. At no time should you be asked to relinquish the bunny, though they may ask you to take your hands off it so they can make sure you are not concealing anything.
If at any time you feel the screener is unsure or requests something you are not comfortable with, you may request to see a supervisor, and they must comply. The Supervisor will have the final word on any procedures. If your bunny gets nevous, you may want to accept a "private screening", where you, your bunny, a screener and a supervisor will go into an area out of view of the public while the screening takes place. This might be quieter and less stressful for your bunny.
In the course of my training, I had to learn how to pat down a dog, because a service animal’s harness will set off the alarmand you need to pat them down to make sure there is nothing else hidden in their fur. There are several pages of guidelines in the SOP about how to properly screen a service monkey, and I once tested a screener who had once needed to pat down a service pony.