I had this same issue when I had Neil Jones build my dies for my 338 LAI. Yes, reaming out your die will work but you my need to ream the seater die out too. I do not know what dies you have?
I did it the easy way
. I made a tool for shell holder and put a redding bushing in a collet. I used a Bridgeport vertical mill to neck size the case. I used this same setup to seat the bullets. I simply took the seater out of my die and put it in a collet and seated them all.
Either way you will need a machine shop to get you rolling, unless you have access to one or have your own.