Welcome to ATP.
I think the first problem with your plan is that Pistache trees are grafted. If you take a root cutting, you will be growing whatever was used for root stock.
You could try grafting a cutting onto another closely related tree. If this tree produces seed, you could possibly grow your own rootstock.
Or you couldd try growing it from a cutting. According to my propagation guide, the best time to do that is mid summer with softwood cuttings.
Daisy
