I’m afraid I don’t understand your question (and I don’t think others do either). I read your post 3 times; it’s lacking some details (understandable, given the frustration).
Tires get punctured all the time. If you’ve never had it happen, you’re in the minority. If the ‘dealer’ you bought the tires from want to charge you for ‘tire repair’ for a puncture, that’s typical of a dealership. Tire ‘chains’ (Les Schwab, America’s Tire, Tire Factory) typically will all perform free flat repair to tires they sell. Dealerships? Well, perhaps a minority, but not atypical.
If they’re telling you it ‘has to be replaced’, that would be due to sidewall damage (cannot be repaired by law). If that’s the case, it’s a simple matter of them showing you the damage. It happens.
Contractors are the biggest culprits for road debris/tire hazards; but regardless, I think if the prior is the case (they want to charge you for a ‘tire repair’), I think you should consider getting a referral to a local tire shop (Costco, even?) rather than doing the Dealer (a Ford dealer, if I understood the vagueness in your post correctly).
Tire repair shouldn’t run more than $10-20 depending on the shop. Getting there, however, is a different story when you have a flat (road hazard helps there).
Hope that helps.
We had almost the exact thing happen to us, at a local tire dealer. He tried to get out of fixing/repairing what we knew was a defective tire.
Call BF Goodrich’s customer service department. Let them know all the things you’ve posted here (pop, rear tire vs front, if it were a puncture wouldn’t it a) be visible/have evidence of some sort and b) tend to leak over time). Also, let them know you see this as a SAFETY issue and that you’d not of expected this from their tire. It doesn’t hurt to try.
Good luck.
Check the date code on your new tires to make sure they didn’t sell you old tires.
Happened to me once a few years ago. I purchased what I thought were new tires and while the tires had never been used - they had languished in the dealer’s inventory for 10 years. I raised a stink and they did replace them with current inventory.
I’m on my 3rd set of BF Goodrich All Terrains...never had an issue.
I try to buy Michelin Tires or not Union shop tires. The Union tire shops seem to have more failures.