I like the idea of doing some test runs. Perhaps ask the lady if you can use parts of the shirts in question for a test run to see if they will work with the tricot. It's nylon tricot, so obviously nylon can stand some heat. I'm thinking the fusible may work okay for the shirts in question as long as you are careful to use a pressing sheet and iron from above the nylon tricot (instead of the shirt side).
Another possibility would be to spray baste the shirts to the tricot instead of fusing with heat.