I believe the tough part here is that, if it's a rimless case chamber, the head of 7mm brass is probably .470"+. A small base die isn't likely to help wherever the shell holder stops the sizing die. Might also want to keep an eye out for where the shoulder ends up if one were to try running a case deeper into a die.
It may be easier to start with brass that has a head/rim size a bit closer to .453". A thought that I know can work is pushing a case all the way through a die, instead of retracting it. Maybe, there's a carbide pistol die laying around, like in .44 mag., that the top can be lopped off. I'd think it wouldn't take much to dig up something to use as a spacer on top of the reloading press ram to push a case past the carbide sizer insert. Only thoughts, I'd definitely try to get it shooting as is.
