I can't see why the tyres would need to ever be weighed. What can be identified from the weight of a tyre? Any modification to try to cheat, would only be detrimental to the level of grip provided in terms of weight. Nobody would wear them down to ridiculous levels as it would affect ride height and plank wear and would become a safety risk. You'd also be dog slow with no grip.
On that basis, surely weighing the car without tyres would be as simple as driving the car over a platform which aligns with the plank, which is able to lift and weigh the car by rising up onto the plank only, like an undercar jack/trolley they put the cars on. Your team are ordered to come to the top of the pitlane and remove the tyres with the provided standard issue pit gun per wheel. Then the car gets weighed with no tyres.
We don't have to be overly reactive given this has happened apparently twice in 30 years (as someone said above?) but it does seem a shame to be so marginal and require pickup to bring a car home above weight. A better system would be to take tyres out of it to allow high deg and pushing tyre envelopes for more exciting, contrasting strategies.