For reasons not relevant here, I made another clamp for a magnifying desk lamp and mailed it off in a small box. A few measurements suggested all such lamps share a common design and similar parts, so I duplicated my previous attempt, with some improvements.
On the upside, the same scrap of aluminum plate I used for the previous clamp emerged from the stockpile and, after a session with Mr Disk Sander, sported two square & reasonably perpendicular sides:

Rather than rely on my original dimension scribble, I transfer-punched the hole location from my as-built clamp to the stock:

That’s a reenactment based on a true story: the actual punching happened on the bench vise’s anvil surface, with too many moving pieces supported & aligned by an insufficient number of hands.
Drilling the 5/16 inch hole required mounting the Greater Chuck on an MT1 taper adapter for the Sherline:

It’s normally on an MT2 adapter for the mini-lathe tailstock, where it handles drills up to 3/8 inch. For the record, the Sherline’s Lesser Check tops out at 1/4 inch and the Least Chuck at 5/32 inch.
Punch & drill the 4 mm cross hole for the clamping screw:

Grab the plate in a toolmaker’s vise, set up some casual guidance, and bandsaw right down the middle:

Bandsaw the outline to free the two halves from the stock, then clean up their perimeter:

Saw the clamp clearance almost all the way through to leave a protrusion, then file the scarred kerf more-or-less flat:

Do a trial fit in my lamp, which lacks the fancy brushed-metal finish of the remote one:

It holds tight and rotates well, so break the edges and shine up the outside to a used-car finish (“high polish over deep scratches”):

The inside remains gritty to improve traction on the lamp stem:

Declare victory, box it up, and away it goes!