If it's cheaper to produce a small car because of the tax, then the tax is effective. Making the bigger cars more expensive incentivizes the smaller cars.
Taxes, fines, and regulatory fees in economic theory are supposed to represent the costs incurred by the general public (in this case the environment as well as infrastructure maintenance) being paid by the parties responsible. This often is not the case in practicality, such as the costs to reverse methane emissions not being covered by the fines associated with flare stacks.
If the companies can't produce cars cheap enough then they'll have to raise the price. If less people can afford cars, that's fine, then more investment will have to be made into public transport, bike lanes, and walkable communities. I do not see any downsides to a tax on larger vehicles.