Welcome to one of the oldest problems in the world- mapping a map onto a globe 
Long story short is that there’s no good way to do this without dissecting your circular texture. Slightly longer story- go look up the different globe projections and learn about the limitations of each. Fact is, there’s no way to map a 2D texture onto a sphere without some stretching- be it a circle, rectangle, whatever, you’re going to get stretching.
My favorite visual aid for this:
All of these circles are the same size geographically. In this case, the stretching is going the other way- from sphere to texture- but the mathematical causes and distortions are exactly the same.
In this particular case, your best bet would be to slice the circle up into a lobed projection - something like this:
If this isn’t an option, then I would take what you already have, bake it onto a new image, and then clean up the stretched edges with the clone brush in Photoshop