On my old FJR, same size tire/rim, I picked up a nail. I decided to use a combo patch/plug however I needed the tire removed from the rim and shortly thereafter, re-installed. I brought my cordless drill and tools. I had my local shop remove the tire, I did my repair, they re-installed the tire and re-balanced. My local shop charges for a new tire if purchased there, then a separate charge for mount and balance. They charged me JUST THE MOUNT AND BALANCE.
IMHO, besides my plug, I think it cost me $12.00. $12.00 is cheap insurance as far as I'm concerned...
You likely don't need a new tire, however you may need a new shop to R&R that tire (I'd insist and pay them to R&R as if it was a new tire or go elsewhere).
IMHO, YOU HEARD IT IN THERE, it didn't go away........ Its YOUR LIFE SHOULD that tire blow, Not the mechanics...