If I wanted a VAT, I'd move to Europe. I don't see how VAT would solve illegal immigration, except to make it more palatable for you. To stop illegal immigration, you need to enforce the laws we already have, and stop giving them food stamps, welfare, under the table jobs, health care, and in-state tuition. The government has plenty of money, giving them more will just give them license to waste more. Nothing knee jerk about that, it's been that way for years now.
There is nothing outside the box with respect to your proposal. In fact I think it is more like head in the sand.
If you are a rocket scientist, then I can understand how launches fail when rockets are loaded with the wrong version of software, or burn up in the atmosphere when metric and American units are used interchangeably. Either that or scientific and political intelligence are mutually exclusive in your case.