When you use a carrot it is faster. The rabbits will be attracted by the trap immediatly and you can literally stand next to it and wait for the rabbit to go into the trap.
If you do it without carrot, then it takes until the rabbit takes the trap. I have noticed that the amount of time is random though. Sometimes 5 minutes, sometimes I have to wait to the other day till the trap catches anything.
So you have to choose a method depending on your current overall food situation.
As far as I know, Rabbits are not attracted by berries and I'd use them for cooking.
As you noticed, 1 Rabbit does not fill much hunger, that is why I always build a cooking pot as fast as I can. As a stew the rabbits are much more useful, or to feed off pigmen so they follow and fight for you.