Perhaps it is just my scientific background but i like a good mystery.
Here is how I would at least start.
1. Measure the expander ball and see what nominal caliber you have. ie, 22, 30 45, etc.
2. Measure the internal diameter of the base of the size die-sit down with a reload manual and match up external case diameter with caliber.
This will at least get you started in a family of cartridges.
Once there, dig out a pile of fired brass and lube the beejeezus out of it and GENTLY see how far things proceed-probably would do this without an expander ball initially so as to easily knock out the case in case things get a bit sticky.
OR--buy or borrow some Cerrosafe and just do a casting but where is the fun in that!
Cheers,
Gary