Having ridden similar pedals forever I have a suggestion - run the strap through the openings in the plate. So under the outside body (as it runs now), then, if you're looking at the picture, run the strap to the right, through the left opening, in the right opening, then through the body and up. It'll be tight but it works. I used to run my straps like that when I had two straps on each pedal.

This will keep the strap from moving laterally, keep it from hanging too far down down under the pedal, and prevent it from going too far forward under the body. The opening when you clip in will be slightly different than what you're used to but it'll work.

Two small zip ties (crossed over the strap so it doesn't push it one way or another) should keep the strap from hanging too, but the cage method will work with no extra bits.