Shaping a JPG image of an externally created logo

I need to bend the JPG image for the logo above into a curve for insertion into an oval label that I will place on the back of a 3 inch button pin.

The curve will sit to the top of the label, making room for general info and contact info.

Everything I read seems to want to help, but when I try to do it, I get nothing to work with.

Where do I start, or is it even possible.