You don't need to use fancy hinges to fix the problem. The admittedly crude diagram, below, is a cross section of the box and lid (the dot on the right is the hinge). The arc is the path of the inside edge of the front of the lid.
The rectangular white space inside the lid is where contents can fit when the lid is closed. But the arc when the lid opens and closes chops off the corner of the contents area, which is the problem.
You can fix it by removing a wedge of material from the inside edge of the lid. Use a little geometry, or stick a block of something inside the box or tray that just fits in height when the lid is closed. With the lid open, push the block against the front of the box and close the lid. You can use a material like carbon paper so the inside edge of the lid marks the block where it hits.
Add a small margin for clearance and transfer that dimension to the front bottom edge of the lid, measuring from the inside edge. The wedge to remove goes from there to the top inside corner.
The wedge can be removed with a Dremel tool, chisel, carving knife, or other suitable tool. Not a lot of material will need to be removed.
This approximation should be close enough. But if you really want it perfect, calculate a larger concentric arc that passes through the inside top corner of the lid. It will intersect the bottom edge of the lid at the same place. As long as you remove the lid material on the inside face of that line (regardless of the actual shape you cut out), the lid will be guaranteed to clear. You can start with a simple wedge, then shape it to the arc. You can use the same test block to guide your cutting; gradually rough out the cutaway until nothing hits the block.
Note that the drawings aren't to scale, they're exaggerated to illustrate the concept. The actual dimensions of the case would involve much less material removal than implied here.
Update: As Matt's comment points out, the above approach would be difficult to make it look professional without the right tools and practice. Here's a much simple approach.
Fasten the hinge with the pivot centered on the inside height of the lid. The problematic bottom front inside edge of the lid will then be a distance from the hinge pivot that is a diagonal. As you open the lid, that edge will swing out because the diagonal distance is longer than the horizontal distance. By the time the edge is at the maximum possible height of the contents, it will be at the original position of the top inside corner. So opening the lid clears anything that might be inside.
This solution does create a somewhat similar problem at the back of the lid, but it's much easier to solve there. As the lid opens, the bottom back inside edge of the lid rotates into the content area. The tiny arc shown on the diagram at the hinge illustrates the points that would be equidistant from the hinge. What is farther from the hinge than that arc is what will rotate into the content area.
This is a much smaller amount of "problem" material to remove, and in this location, all it takes is some handheld sandpaper to round off the corner.
Keep in mind that the diagram isn't proportional to the actual lid dimensions or wood thickness, so it illustrates the concept but doesn't accurately represent the amount of material that would need to be removed.