I recently bought a '16 Cooper without a front license plate holder. I bought one and when installing it the screws stripped in the plastic bumber before they even began to feel snug. So I bought some thicker screws, and they did the same thing. It's as if the plastic is extremely soft and no screw will get any bite. is there a fastener people have used here that's effective? I'm thinking along the lines of an expanding anchor like one would use in drywall, only shorter.

 

Double sided 3M heavy duty tape and maybe try that anchor. I seen some guys just get rid of it and fix the bumper holes and go with a tow hook plate mount:

You can build your own here.

https://www.ecstuning.com/b-ecs-part...AAAAAAAAYBAgID

 

On my son's Subaru, I used stainless bolts and nuts with fender washers on the backside. It's very secure. Stainless costs a little more but I hate rusty license plate screws.