Bonus points if you make the fee scale by weight to the fourth power, to match the relative impact it has on road wear

Rego in NSW does scale by weight. Other states should bring that in if they don't already. However, that hasn't stopped yank tanks from flooding NSW roads. They should also require an LR licence since they are literally the size of a truck.

It should be exponential too. That's how road wear works, so make them pay in accordance to the damage it does to the roads.