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There is going to be a bending action there at the L-bracket's pivot point. It being a problem should become apparent with trial and error. Perhaps bending the pivot bolt upwards by about 20 or 30 degrees might strike a balance in terms of the pushing force on the door. Or raising the pivot point as far as possible from the door will also lower the bending action enough to let it work (invert the L-bracket to have the pivot point even higher?). A spring might also come in handy to counteract the weight of the door although the door lock is probably plenty strong.
Best of luck with this. I tried this with bike brake cable but it didn't work out.
ollie
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