different case lengths
One thing I haven't seen mentioned is the difference in nominal case lengths,
.308 win, 2.015,-- 7-08, 2.035.
This means that when necking a .308 down, you end up with a 7-08 case that's .020 short. No big deal, just be aware that neck tension will be less,, because of a shorter neck, but may be compensated for by the neck being thicker. Also trimming to square up the mouth, could only be done with a trimmer adjustable to less than minimum trim-to-length,(that leaves out the lee system).
A better bet is necking UP from .243,( .243, 2.045), would result in a case that's .010 LONGER than a 7-08 should be. Also the necks would be a mite thinner than normal 7-08 brass.
As has been said, 7-08 brass is inexpensive, plentiful, and the headstamp is correct!