You can push the 7x57 deep(er) into a standard 30-06 FL die (if you happen to have one around) and not set the 7x57 shoulder back.
Using a LR Primer Pocket Swage replacement for the standard shellholder makes a nice pusher for the case in this process. It is flat, allows the case to be pushed as deep as possible into the die and the p/pocket swage punch keeps the p/pocket on the case from deforming/downsizing as a result of the solid base swage work.
A suitable sized steel washer is of course much more likly to be on hand in most reloaders shops and works well too. Just check the primer pocket after the work and make sure they didn't reduce in size a bit. The matr'l has to go somewhere.
Using the FL die you can swage the solid base down but you need a very strong linkage advantage press to do it and only in very small increments of a couple .000 at a time.
Don't skimp on the case lube and use a knock out pin to punch or press (prefered) the swaged case back out.
Don't try to use the standard shell holder/case rim method to withdraw it when doing this. It'll shear off just about every time.
You may have luck using a 45acp FL sizer also. They are usually about .467/468 at the base.
Plus no issue with the case shoulder above.
A carbide sizer die may crack the carbide 'ring' inset into the steel die body if you use it for this swage process.
I'd use a standard old steel FL sizer for the try.
