What you have is a trip on in-rush current. I would venture to guess you have a 600A breaker on the 208V side, and a 225A breaker on the 480V side. Using a standard damage curve and showing inrush current, both breakers are going to trip on inrush. Which one, is a race, but I'll bet it's the breaker on the 480V side.
You need to get the manufacturers damage curve for the transformer, and inrush multiplier. Then select the protection such that it will not trip on inrush but also protects the transformer from damage. I would investigate using a 500A breaker on the 208V side, that has an adjustable instantaneous trip to get you out beyond the inrush. On the 480V side you are probably going to have to resort to time delay fuses.
I am not surprised by this since IMO the NEC charts work poorly for properly selecting protection of transformers.