22 is too thin. I know it's a short run and all, but I like to get people in the habit of doings things right the first time.
We can look to the wire used on the actual transformer for a good guide on how thick it should be.
You know if Williams could have saved some money and used thinner wire, they would have....
An old ATX computer/power supply is a treasure trove of stranded wires of many colors.