Hiya,
I've not tried it myself, but don't see why it wouldn't work.
I've seen kits for kids where carrots, leeks etc are grown in big test tubes to let them watch - so obviously carrots are capable of growing in a restricted space.
On the subject of your destroyed carrots - was it pest damage that destroyed them do you think or the actual pulling of them out the ground ? Heavily manured ground that's not had time to settle is bad for causing forks in carrots, perhaps it's been something like that or our friend the slug (evil little blighters!) in which case they might be able to navigate your bottle and get the carrot anyway.
I'm sure you'll get plenty more advice from people more expert than me though- good luck and let us know how it goes.
Karen