I would probably start by creating a checker pattern in Photoshop, if there's not already one in the presets. Then distort it to match the projection, using the method of your choice. Then intersect a couple of rectangular selections to create a mask the width and shape of the desired border and mask out the checker pattern with it. Then it's a simple matter of carefully adjusting the checker pattern so that it fits within the mask properly. If you unlink the mask from the layer, you can slide the pattern around underneath the mask.

Just be sure to make your checker pattern oversized so you don't lose too much resolution during the warp. That should keep the edges good and clean.

Since you've got kind of a spherical warp there, it might be difficult to keep the verticals where you want them to go. In that case, it might be a good idea to make two layers: one for the horizontal borders and one for the verticals. Prior to doing the warp, resize the checkers so that you have only alternating horizontal lines in the vertical layer and only vertical alternating lines for the horizontal layer. Since the pattern is orthogonal, you shouldn't have any problems with filtering softening the lines when you stretch it out like that. Just in case, though, you might change the interpolation method to nearest neighbor before making that transformation. Just make sure you change it back before you do the warp or you'll wind up with horribly aliased lines.