I got out my calipers and the ID at the bottom of the sizing die was about 0.01" narrower than the O.D. (0.473") of the casing.
Did you measure the outside diameter of the casing, or did you use the specified diameter found in reloading manuals?
I just measure some .308 brass new and fired. The new stuff is mic'ing out to .464" at the casehead although the specs given in my reloading manual is that its supposed to be .473" I have a set of .308 small base dies that measure out to .463" at the base. Fired brass is running at .467-.471" right above the casehead depending on the brand of brass and if I run them through my small base die versus a FL die, it takes quite a bit of force.
Unless I'm mistaken, the 7mm-08 brass should have the same casehead diameter as a .308.
The dies are made undersize because the brass is springy- if you squeaze it down, it will spring back to a certain extent. I observed this while trying to set a consistant cartridge headspace for getting the maximum caselife of '06 cases out of my garand. I could only bump the shoulders back so far reliably- if a case happened to be .002" longer in headspace than the rest of the batch, the shoulder wouldn't bump back to the same length as the other cases, because it would spring back that much more.
I would remove the decapping/expander ball assembly from the die and run a well lubed case into it. It will take some force considering you have a tight die similar to a RCBS small base sizer.(I mostly use RCBS FL dies which are looser so they dont' work the brass unnecessarily.) If this works, put the expander ball/decapper back in. adjust it so that you can just see the pin go all the way through the shell holder with the ram all the way up, and try again. Don't be afraid to lean on that press handle.