Exempting one form of heating and not another seems wrong (especially when the exempted one is "dirtier" as far as I know). Is Canada not redistributing the carbon tax? Many other countries do it that way - all or at least a significant part of the carbon tax is paid out to the population, so if your consumption is below average, you get more money than you pay.
That way, people remain able to afford things, people who consume less CO2 intensive products than the average are actually better off, and the incentive to optimize for reduced emissions remains.