Hi PH.
I used an old FL Pacific .222 die. I measured it up, finding the body to be .375" which would fully support a .375 bullet's sides to prevent expansion of the bullet's jacket while travelling through the choke.
I turned the die off on my little lathe in the shoulder section, about 1/2" above the start of the neck. I then chucked the die in the 3-jaw and ran the tool into it on a V slope until I had a diameter on the shoulder angle that measured about .355".
I then used a .375 neck/throating reamer to expand that dia. to .366" and tested it using a 225gr. .375" bullet.
I screwed the die up into the Orange Crusher Lyman press, put a Lee post of .350" in the ram.
I lubed the bullet with Imperial die wax and pushed it through the die. It came out at .364" IIRC. I then, with my fingers, carefully turned the neck/throater into the die again and opened it up a bit more, testing as I went, and stopped when the bullets came out at .367", the largest size my rifle would allow to be chambered and released from the case.
I wrote this up here(I think), as well as at gun boards forum and a fellow from California sent me a set of RCBS .223 dies to do the same with. I don't remember which forum he was from.
1st time lucky, I guess. The FL die didn't work at all and the seater die did, after a fashion but seems to me it took a lot more force than my die, to push a bullet through. I had to anneal those dies, but not my old Pacific die for some reason.
With my die, the force needed is about like sizing a .300 Win Mag case.
No, it isn't for sale.
Since that time, we have learned that Lee Precision will make a set of 3 sizing dies to do the same trick my one die does. They use their cast bullet sizing dies that fit a press and with their angles, ect. 3 dies are necessary, at about (some time back) $30.00 each.
I found that lubed .375 cast bullets size down perfectly to .367" without any distortion as well, whether lubed with Rooster Red or Lyman's Moly lube or 50/50 BW/Alox.

A buddy of mine up here bought a couple sets of Lee dies and makes .330" bullets out of .338's.
I think he also has a set of these Lee dies to make .315" out of .323" bullets.