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Your items 1 & 2 alone would solve the problem:

“1) Secure our borders
2) Enforce existing law”

If no additional illegals come into the country, that would stop the flow, which is the real problem.

If no additional illegals come into the country, that would stop the flow, which is the real problem.

The border's total length is 1,954 mi, according to figures given by the International Boundary and Water Commission. It is the most frequently crossed international border in the world, with approximately 350 million legal crossings being made annually.
[…]
There are an estimated half a million illegal entries into the United States each year.

Wikipedia on the Paladin:

Crew:                     6
Effective firing range:  11 mi
Operational range:      216 mi

So to cover the border we'd need 1,954 / 22 ≈ 89 Paladins… but that's an under-count because that would entail butting them directly on the border. We would actually want them back a bit.

c² = b² + a²
11²  = 5.5² + a²
121 = 30.25 + a²
90.75 = a²
9.5 ≈ a

So, we could put them 5.5 mi away from the border, allowing them to cover 19 mi of border; this would mean that we'd need 1,954 / 19 ≈ 103 paladins [103 * 6 = 618 men].
Or, we could put them 9.5 mi away from the border, allowing them to cover 11 mi of border; this would mean that we'd need 1,954 / 11 ≈ 178 paladins [178 * 6 = 1,068 men].
(Of course we'd need logistics to support them: meals, munitions, mail, etc.)

See, securing the border is not as hard as they make it out to be.