bavarianbrit,
The .318" diameter portion of the hole doesn't need to be very long, 3/8" is plenty, otherwise there will be a lot of friction involved. Also, you didn't say how you tried to push the jacketed bullets through the die. Your neighbor can make a "pusher" for you with the bottom duplicating the shape of a shell holder and with a hardened rod (less than .318" diameter) attached to it to push the bullet through the die. Lanolin based lube works well and can be found in the Baby section of your local sore (used to prevent or threat diaper rash). It is easiest applied by dipping one of the bullets into the lube to get a blob of it onto the bulled. This bullet and the rest of the box are dropped into a one-pound size coffee (or other) can and swirled (for lack of a better word) around to lube all the bullets at one time. After the bullets are sized, the extra lube can be cleaned off with lighter fluid on a rag or old towel (acetone or alcohol evaporate too quickly). You cannot size solid metal or partition bullets this way, they have to be "cup and core" bullets. Also, the press has to be heavy enough to avoid damage. You can have your neighbor bore the top of the die to a diameter more than .318" to within 3/8" of the transition. You are on the right track, good luck. In addition to S&B, you might try PRVI for .318" bullets.
Mike
