I would buy another brand of die to start with, like a Redding FL die for starters. Not to get into a 'which die is best' ****ing match, but Lee usually does not get mentioned in 'best' discussions. Lee usually gets mentioned in 'cheapest' discussions.
Also, this was asked earlier in this thread, but will a once fired & twice fired brass fit in the chamber before sizing?
Could this 'bulging' be happening during the FL sizing.
The Redding Competition Shell Holder set allows you to keep your shell holder / die contact cam over, while reducing the amount of shoulder set back in .002" increments, depending on which shell holder you use in the set.
He is ALREADY using Redding and has tried Lee as an alternate. The issue is NOT the make of the die it is that his chamber is smaller than Saami spec.and a standard FL die of any make is not designed to size brass for below Saami spec chambers.