Doppleganger,
Very cool, that's along the lines I was thinking but with a bent edge flange to increase rigidity. Something like this.
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However, the flange is gonna have to be pretty short. I though there was more room between the panel and the window because when the window is down there is plenty. However, as Squareroot pointed out, when the window is up there is barely any clearance with the supports at the bottom of the window.
I also noticed the existing reinforcement in the door area there isn't bonded to the panel out at the edges:
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So I can slip the stiffener in a little way but not all the way to the screw. I think bonding the existing support out at the edges as well as using something like you did or something with a small edge flange might be enough.