Is it not a reasonable interpretation of this clause that the United States Government has the right to regulate its borders?

The federal government does have the right to regulate its borders, but it comes from other sections of the Constitution than that Clause. The Invasion Clause certainly applies only to protection from foreign armed forces, and wouldn't apply to civilian migration. We know this because the Founders never imposed any restrictions on migration, just nationalization. For the first hundred years of our history, any civilian could cross any border at any time for pretty much any reason - if you wanted to walk into the country and live here, you were allowed to. That didn't change until 1882, with the Chinese Exclusion Act. So it's pretty clear that the Invasion Clause referred only to military forces, not civilians looking to move to (or just travel in) the U.S.

And don't confuse the government's right to regulate the borders with an argument that it is somehow unconstitutional for them to adopt any particular level of migration regulation, even a very low level of regulation if they chose. If the U.S. government decided to go back to the pre-1882 regime of having no controls on civilian migration, it would be bad policy - but it wouldn't be unconstitutional. So there's no constitutional limitation on the federal government agreeing to a treaty that provides basic rights (and pretty minimal ones) for refugees - they can commit by treaty to doing anything they have the power to do without a treaty. By way of example, the Congress has the power to set tariffs - which includes the power to have no (or very low) tariffs - so even though the have the power to regulate trade, they always have the ability to enter into free trade agreements and other treaties where they agree to not exercise that power (or exercise it very lightly and according to agreed-upon rules).