I have an old 6.5 Japanese that I've had for decades. It has the same problem. I found if I did the tape trick on the first firing the bulge was more concentric and en empty would rechamber, but the bulge was still so bad there was no real way to neck size only.
The way I got around it was one day at a gun show I found an old set of "6.5/308 Win" RCBS custom dies for $10. Adjusted properly, these worked fine to neck size.
I made a few 6.5s out of 220 brass back in the day, before obscure brass was as readily available as it is now. The only problem with the converted brass is that it bulges just the same.220 Swift brass (if you can find it) can be made into 6.5 Jap