I have a 2.5 year old girl that can be described exactely like your dck. She comes here with her 'binkie' (bunny blanket). She carries it around with her everywhere. I've noticed that she drops it and leaves it behind. When she notices it is not around, she gets very upset. We have to hold a search party for it! lol. I don't think she really needs it anymore. I don't mind giving 'binkie' to her at nap time, but I don't think she needs to have it with her all the time. I've tried putting it in a bag to get ready for home, but she notices it out of sight. It took her a long time to get used to being here (since she is PT only), so I think she needed 'binkie' in the begining. Lately, she leaves it behind a little more.
I am with you on only giving it to her at nap times. My dck will sometimes get very upset when I tell her she can only have it to sleep with, but eventually calms down when she decides it's easier to play with other toys when she is not lugging it around. lol
Good luck!