That looks like a 1 culm fargesia division of some sort. I think you probably should of started with a much stronger plant since small divisions are prone to things like this. At this size you could use a wall o water to prevent bunnies from chewing up and give it a chance to put on some size and save it.

Wall o water will also speed up the growth rate a bit and help get your little boo established.

Here is a link that might be useful: wall o water

Ever tried hassenpeffer? It's really good. Little bunnies are very tender.

I had this problem for years with squirrels. The only solution that I found to work over the long term is some kind of hardware cloth or heavy gauge wire mesh that can be formed into tubes and set around the new shoots until they are at least 18" tall, or so, and have hardened off. These protective tubes can be attached to other culms or to stakes to keep them in place.