Hardly a "freak"! You had one squib that failed to clear the barrel. You didn't recognize it for some reason and proceeded to fire several rounds after the squib. Because of the bullet already stuck in the bore the next one stopped when it struck the stuck bullet and subsequent rounds just piled up until you finally realized something wasn't right! What the reason is for the squib is not important. It could be a light charge or poor ignition of the charge due to a weak primer or simply cold weather. There is a good chance the first bullet would have cleared the bore if it had been lubricated lead instead of jacketed! YES, the coefficient of friction for plated bullets is the same as if they were a conventional cup-and-core bullet.
There was only one reason for the incident, and that was the squib. No tight bore! If you fire a round and either it doesn't feel right, or sound right, then STOP shooting and see what the problem is before firing even a single other round of that ammunition. You cannot blame this on anything except yourself.
SAAMI .38 Special standard bore diameter is .346" to.350", just like Uberti states. The "tight bore" was just an excuse!
$700 seems quite exorbitant to simply replace a bulged barrel. I would expect no more that half that amount.
There is no reason to be concerned about shooting the gun once repaired, just pay attention to what is happening! Unfortunately you aren't the first to have this happen. I bet any gunsmith who has been in the business any time at all has seen a situation exactly like this come in the door.