Don't do it!
I can speak from painful personal experience!
I shoot a lot of 308's and bought a 7-08 remington bolt figuring I could do the same thing. I tried it and no problem, brass chambered and I trimmed to the same length as standard 7-08 brass.
I next (getting a bit cocky) loaded my brass up with 130 grn Seirra JHP's and I started 10% off the Seirra Max load. (I do not want to give Seirra a bad name, so no other details).
First shot my hat came off, my muffs were pulled back and I was glad I had shooting glasses. I blew the primer right out of the winchester brass and took a good blast of gas and powder right in the face (actually as I reread this it was a big blast of gas). It jamed my firing pin in the bolt, and I could barely yank the action open without tools. The primer was gone and the whole base of the case was blackened. The bolt face was lightly scored and I had to break the bolt down to get the ejector and firing pin to work again. I was lucky to have my sight. (Always, and I mean Always wear shooting glasses)
I credit my typing this to you because of good remington rifles and starting 10% down.
I know it sounds like a good idea, but I store this case on my reloading desk to remind me of my stupidity. Actually it probably made me a much better reloader because I follow every rule now and document everything.
ps: admitting my reloading stupidity is good for the soul, but I get shivers when I think about how close I came to ...