Why on earth are YOU ordering the parts and paying the machine shop. That is the repair shops job. Now it's just a case of who shot John. If the engine is bad, the repair shop can't go back to the machine shop because the repair shop didn't contract for the work. And the machine shop isn't going to deal with the repair shop because they didn't pay for the work.
To be clear on "paying" the machine shop-the shop I'm using dealt with them directly and paid them. I paid the shop right after they'd paid the machine shop.
Those particular parts that I supplied never actually passed through my hands-I had them shipped directly to my mechanic per his shopping list. It wouldn't normally be done that way, but he was willing to do it since at the time Moss Motors and the like weren't open and shipping parts.