Had the same issue and don't want to buy a full set. I just ordered one from amazon will update this post when I see if it fits.

Any luck with the socket you purchased? If so can you please share a link to it?

I received the “

Stahlwille 01010012 Steel Hexagon #40, Gentle, Resilient, High-Performance Steel, Chrome Plated, Size 12 mm, Length 23 mm, Width 16.5 mm”

from Amazon.  It works perfect for the Vector 3 pedal spindle nut.

Try searching for "thin wall" sockets.  They exist and are designed for this type application.

I ground down a Craftsman 12mm 1/4-inch socket a bit on a bench grinder. Worked like a charm. Tip from guy at my local bike shop, Cycle Progression. P.S. I had a million 12mm sockets 3/8 and 1/4, deep and short, and not one of them worked. The grinding was the bomb tho.