The US does export some Alaskan oil
It did...temporarily.
When an oil glut developed on the West Coast during the Clinton administration, a bill was passed allowing the export of some Alaskan oil.
However, the decline in Alaskan oil production has eliminated the glut and the West Coast oil refineries are having to "import" Bakken crude to fulfill their needs.
At this point, I don't think any Alaskan crude is being exported.
From 1996 to 2000, about ~5.5% of the oil produced in Alaska was exported during the West Coast oil glut. Since then nearly no Alaskan oil exports.