Savage did use a different mag box and follower for the 7RM and the 300WM and other belted mags in the SF actions. However some people have had decent results using 30-06 based cartridges in the magnum box/follower and vice-versa. Give it a try and see. You may also be able to tweak the lips of your 270 mag box to feed the 300WM.
Only way to know for sure is to try it, but switching to the 300WM box/follower would be a sure a thing.
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