Convincing a king might be easier. And as I point out above, the support of a majority of Americans hasn't helped the cause of immigration reform.
You would, in essence, prefer a tyranny, as long as that tyrant agreed with you.
Since I do not equate Monarchy with Tyranny, and in fact believe that historically Tyranny has generally been the consequence of abolishing Monarchies, no, I would not.
An American Catholic monarch would allow unlimited immigration, legal and illegal, since that is now, and has always been, the position of the Church.