I'm having a hard time visualizing where the wires and box placement is with regard to the shutter, but how about cutting a hole in the siding out the so you can put one of the siding light mounting blocks. Rotozip or one of the oscillating saws (fein clones) should do the job OK...

Here is a link that might be useful: What I'm talking about.

The house is brick. No vinyl siding. The 2 x 3 horizontal box was installed as the brick were being laid. The front of the box is flush with the front face of the brick. The location chosen for the shutter has the box (in the brick) offset from the center of the shutter. The shutter is installed over the box.

AH, so it's not getting the thing shimmed out to the face of the shutter, it has to move over a couple of inches, that's going to be a bit more challenging.

If I were faced with the same situation around my own old house......I'd use two pieces of solid sheet 1/2" PVC to construct a two piece custom mounting bracket......

Permanently remove enough original shutter slats to expose the existing box and Tall/high enough to visually support the new fixture...size the PVC blocks to fill the space exactly....

The bottom BASE gets a grooved wire chase...and a 2x4 hole around the existing box.....mount the base to the brick using a center mounted 1/2" round fixture box (I'd try to use horizontal mounting holes).....align the groove between the boxes....extend the wire....

The COVER gets a 4" hole for the fixture box......and a single small "hole" that aligns to the existing box about an inch away.....(Use a cut-off screw for locating)....add a Trim Washer to avoid countersinking the final screw....I'd re-thread the existing box to accept #8 sized fixture screws.

So......The BASE is firmly attached to the structure....and provides shelter for the wiring....The COVER is removable ...held in place with the Fixture and one Trim screw to the old box....removing the COVER gives full access to all of the conductors and splices.

Being a belt and suspender kind of guy......

I would probably Rip the BASE about 2" smaller..... Glue-Laminating the 1" to each backside of the COVER.... making a U shape that surrounds the BASE

Forming a self-centering/self supporting cover

YMMV

Having a light fixture on top of a shutter is a stupid look and I'd just take off the shutter and mount the fixture to the house where it should be.

I understand what you would do. But this is not your house nor mine. And removing the shutter will leave six holes in the brick beside the front door. We could propose a law about no light fixtures on shutters and hire inspectors to enforce it. Almost everything is already illegal.
I am trying to accomplish what this friend wants within the constraints of existing law and circumstances.
abnorm has offered some very creative suggestions. I am grateful.

I guess if you could move the fixture to the side of the shutter you would consider that possibility.

Sorry, but I have to type it, the thought of a light mounted to a shutter makes me shudder too. I don't even like fake shutters.

"Each to his own taste, said the lady as she kissed her cow".
I suppose that those who make and sell the fake shutters like them.
These are as good as any fakes.

The suggestion of abnorm proved quite valuable. I now know that when the house was built, wooden non-functioning decorative shutters were installed and the present ones are replacements for the wooden ones. The light was always installed on the shutter as the mortar in the screw holes and packed where the screws pass behind the face of the box was still in place. Fearing that the fasteners holding the shutter would break rather than pull out, new fasteners were located and purchased before trying to remove the shutter. Turns out the box in the brick/mortar was positioned better than originally assumed and making the PVC insert was a bit easier than anticipated. Aligning the box to the fixture location required making a template prior to cutting the PVC. Another problem with the made-in-China fixture is that the screws that accept the acorn nuts holding the fixture are 4 x 0.7 metric rather than the 8-32 we are so accustomed to using. With the PVC panel, longer screws are needed and the metric are not easy to find. So retap the mounting bar to 8-32, get acorn nuts in 8-32 and mount the fixture. The customer does not fully realize how much time and work this project has required. But he is pleased with the results. And that is the real bottom line. Thanks for your help.

You're welcome......

Just curious... where did you purchase the 1/2' solid sheet PVC?

Amazon offers several different stock sheet sizes. Not low cost after one adds the shipping. But available. I bought this from a plastics supplier in the nearest city. They cut it to my specs. This is the sintered PVC which has a very low gloss surface. Other types are available. They also had high gloss white acrylic- not good for this application.