What can try it deleting the Update for MW3. Then update it again using multiman;
Start Multiman
Hover over MW3
Press SQUARE
Click on UPDATE
Then let it update the necessary files for you

When it's done it should say that pkg files can be found in Install Package Files folder.
Quit to the XMB, go to that folder and find the latest Update for MW3 which is 1.24.
After that you'll be done
If it still gives you the error, go back to multiman, hover over MW3, press SQUARE, then choose " TEST " and then " SET PERMISSIONS " then try again.
Hope I helped