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Can you do this purely with pressure?. Lift the car off all 4 tires then add a known pressure to all tires, just enough to seal the beads. Then lower the car and measure the pressures again. I'm sure you could get to the ratio of weight distribution for your car. And with the right coefficient for your tire you could probably accurately approximate weight.

if you know the exact volume of the tyres and account for any temperature change this should be acurate

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