For re-bushing requiring the thinnest cloth I have often used Popsicle
sticks or made my own cauls out of wood.

Bring your micrometer to the hobby, art, craft or machinist supply, and
find something the thickness you need.  First measure the diameter of
the balance rail pin, make your caul 2 to 3 thousandths of an inch
larger than the pin, and select a bushing material that makes your caul
fit snugly but not tightly.  In spite of my best or worst efforts, as
the case may be, I have wound up with a dilemma similar to yours.

A company called Pianotek here in the U.S. sells a temperature-
controlled soldering iron that includes brass cauls of many sizes.
This great tool will solve your problem and easily pays for itself.

Bill Maguire