I just had a thought. Well, not just. More like 5 minutes ago, but I had to wait for the headache to subside and the burning smell to go away....
Start forming by running a .222 Remington case (.223 would also work, but I think .222 would be easier to start) through a .218 Bee sizing die. Trim to 1.387" (or thereabouts, 1.314" should give you closer to one caliber of neck if my arithmetic is right). The .218 Bee has a 15 degree shoulder, which is not our goal. .222 and .223 Remington have 23 degree shoulders, which are better, but also not necessarily our goal. A 30 or 35 degree shoulder would be better, and easy enough to fireform to, I would think.
You'd have to get a custom FL sizing die made for the fireformed cases, but if you're custom cutting a chamber, that's not that big a deal, actually, especially if you're buying the reamer.
Not sure what the case capacity would be, exactly, but you'd effectively wind up with a rimless, slightly shortened (mostly in the neck) .218 Bee Improved.