[mothered]

Will this lead to an investigation and a police report before the refund?

An Investigation may be opened with the carrier, but because the (apparent) theft happened In transit, a PR Is not likely to be requested.

Of course, some reps fail to use common sense and ask for the report. 

I have another idea, I have the number of the courier who delivers me, every time I receive a package, I call the courier and I go to meet him, it happens away from home. Could DNA be okay?

It's not good practice to personally target the driver, and use him as bait to satisfy your DNA needs.

SEing must be kept on a non-personal level.