Welcome to the Cake Decorators Q&A

Could the fur on the bunny cake be piped with royal icing rather than buttercream?

Hello Cakeartbylauren

Royal icing dries very hard and won’t be very nice to eat. A softer royal icing can be made by adding 1 teaspoon of glycerin to 450g of icing sugar. You’d need to make a small amount to see whether it will hold up to piping. Alternatively you could use chocolate butter cream which not only tastes great but also pipes really well.
For chocolate buttercream recipe have a peek here http://www.cakeflix.com/using-buttercream-giving-buttercream-more-structure
Colour it as you would ordinary buttercream.

My concern is with buttercream not being able to leave it out. I am in Australia and weather is often warm. Today is 34 degrees!

I know in other hot humid countries high ratio shortening is used in the normal butter cream to help prevent the melting. You can see the shortening here http://www.cakesaroundtown.com.au/catalog/high-ratio-shortening-p-5214.html
For other ideas:
http://www.designmeacake.com/crustingbuttercream.html
http://www.wilton.com/blog/index.php/high-humidity-icing/
http://www.craftsy.com/article/stabilizing-american-buttercream
http://www.craftsy.com/blog/2013/07/crusting-buttercream/
http://thymebombe.com/2012/06/22/crusting-no-melt-buttercream-of-my-dreams-and-my-favorite-vanilla-cake/

There is another one which I can’t find, it contains marshmallows, my search for it has failed. I had it just a few minutes ago and lost my text to you. If you search high ratio butter cream with marshmallow you may be luckier turning it up than me.
Hope the other links help.

Or you could also use strong ganache, use 4:1 ratio white choc to double cream, add colour during the melting. You would need to keep your hands really cool during piping and work quite quickly in an air con environment.