I have two suggestions, one of which can be done entirely within Photoshop.

In Photoshop, assuming you have a recent version that includes the 3d functions:

Step 1: Create an equirectangular graticule. That is, an image that is twice as wide as it is tall including the grid lines that you want to see in your final image.
Step 2: Go to 3d > New Shape from Layer > 3d Panorama. You are now looking at the interior surface of a 3d sphere
Step 3: Select the 3D Pan Tool and dolly backward until you see the viewpoint pass through the front surface of the sphere. Then dolly forward to just inside that surface.
Step 4: Increase your canvas size dramatically until you can see both poles.

You should end up with something like this:

Click image for larger version.  Name: projection.png  Views: 3301  Size: 339.1 KB  ID: 51211

You could even draw your map in the equirectangular projection and use this technique to distort it, although you'll notice that it does introduce some perspective distortion as the image moves away from the center. Things get wider and blurrier as they approach the camera. If I were to use this technique, I'd probably make the graticule and the coastal outline, distort, and then redraw to correct the artifacts.

The other suggestion is to look up a piece of free software called g.projector. It's published by NASA and can convert projections from one to another. I've never used it, but several people in this community have.

Thanks for the mental exercise! Just what I needed to get my Sunday morning started!