Making a circular object straight

The tricky part is that you have to make enough of a gap so that Puppet Warp doesn't interpret the shape as a circle. Here's what my initial selection (that didn't work) that went between the tail and the mouth looked like:

And here's how Puppet Warp got applied:

When I removed more of the tail to create a bigger gap, then Puppet Warp applied along the curved shape:

Then, using Puppet Warp, you place pins along the mesh to control the placement of the mesh. I started by placing the first pin on the neck and added pins about ever large scale or so. 

I'd recommend starting from the pin in the tail. Move it slightly outward to the left. Then select the next pin closest to it and drag it slightly to the left and down. Continue shifting the pins to unfurl the shape. You can move the pins a greater distance, but you may wind up distorting the content.

At any time, you can accept the Puppet Warp if you want to check it. In my case, I accepted it to add more canvas (Image > Canvas Size...).

Then I applied Puppet Warp and moved pins again to continue unwrapping it.

As it got straighter, I accepted the Puppet Warp again and used Free Transform to rotate it.

That's when I got the result that I posted earlier. The sample I made wasn't completely straight, so you would probably need to work on moving the pins a bit more.

Once it's straight, you'll want to include the part of the tail that needed to be remove to make Puppet Warp apply the mesh to the subject and not as circle.