What you're suggesting sounds easy but in order to keep the router flat I would need some sort of support around the neck pocket, no? Then I'd need to make sure the support is flat and 100 percent parallel with the body to prevent the pocket from being uneven. I don't have a planer, so I think this will take me several hours to pull off not to mention the cost of a router, bits, wood, clamps, etc.
You don't need a planer, etc... All you need is a piece of 1/2" MDF to make a larger base for the router, which keeps a lot more surface on the face flat of the body. Making a new baseplate is about a 10 minute job, you'd use the existing plate to mark screws and bit holes, then use your existing screws after countersinking them.
If your bit is long enough, you can attach a new 1/4" base to the existing base with double sided carpet tape.
If you don't have a router, take the body to a luthier or experienced woodworker, they have a bit of a learning curve and good bits aren't cheap.