Since oil companies will make good money from the price hike and exports, I guess they’ll lobby against an export ban?

Oil is not a single good. Different types of crude produce different products at various grades hence why diesel is so expensive in the US compared to something like natural gas

Why do you think Republicans would institute an export ban? Trump has been touting every export deal he can (especially promises to purchase petroleum), he has even been measuring import/export levels as if foreign countries are cheating the U.S. if they export more to the U.S. than they import from it.

I am not confident that the Democrats would even implement such a ban, as they get pretty big campaign donations from the oil industry as well.

About as quickly as the Canadian Oil patch would be flooded with money. All that would do is push oil investments out of the country, and your surplus would become a deficit in the drop of a hat.