The cardboard "template" is made by cutting a PERFECT halfcircle into the edge of a (large) piece of stiff cardboard. You then hold the cannonball fragment's outer surface up against the edge of the cutout in the cardboard, to see if it fits snugly, or is too large or small to snugly fit.
In your case, get three pieces of stiff cardboard, about 12" long by 8" wide. Draw a 6.4"-diameter halfcircle on the long edge of the first piece, a 7"-diameter halfcircle on the second, and an 8"-diameter halfcircle on the third. Use a razorknife/boxcutter to SMOOTHLY-&-NEATLY cut out the halfcircles. Then, use the large pieces of cardboard to check which one your fragment's curvature fits best.
The red lines in your photo are oriented incorrectly for measuring the shellwall's thickness. They need to be oriented as a "Radius line." If you have access to a digital caliper, please use it to check the shellwall's thickness in several places, and tell us what the measurements are. In the photo, the thickness seems to vary a little bit, being slightly thicker on the left end than the right end. If that's true, measure in three places (each end and the middle), add up the numbers, and divide by three, to calculate its average thickness.
You mentioned the "New York Light Artillery." That unit's name indicates it was trained to use Field Artillery cannons (such as 6-Pounder & 12-Pounders), not heavy Artillery cannons (such as 42-Pounders and 8-inchers). Still, I suppose that Light Artillery unit could have been temporarily assigned to some Heavy Artillery cannons.