How to correctly mount double-lock grips?
Short answer, no!

Almost certainly the entire grip should be on the handlebar.

Then the 2 pieces of steel pushed in by the screws will tighten against the handlebar giving much better security.

The steel pieces should go in the cutouts on the hard plastic inner of the grip.

I'm not sure how well the grips will work now that they are missing some of their structure, but it's worth a try.

FYI, the reason that the hard part is more difficult to get on is that the multi part grips can get away with a looser tolerance under the rest of the grip, but not where they are actually attached to the handlebar.