I want to get the tweeter case/basket (not sure the correct name) flush with the outside panel (the one you see)

Here is the back side.

The "basket" has 2 screen holes.

My plan is to get a thin but strong metal bracket (wood would take up too much space), have it hang over each side and put 2 screws or bolts that press against the inside of the door panel (which would be backed with another flat piece of metal to prevent it from poking through the door panel)

Any ideas on how I can go about doing this?

My dad has some nice road signs that would be perfect, but I cannot use those.









As you can see, it looks flush, but it will not stay on it's own because there is no clip in the case itself.

 

That's exactly how most manufacturers do it. Any kind of metal with enough strength to relatively keep it's shape will work. On the other hand, you could always run a bead of hot glue around the tweeter. It will hold it real well and still allow removal in the instance you change sets or vehicles.