Hi Smiddy,
My thoughts . . . ? Well, first, the thread title. To be fair, what you are asking is more of a Bible proposition or problem rather than a JW one.
But, beyond that, the problem, as you stated it is, 'How can the governments of the world be both an arrangement of God and under the rulership of the devil?'
Have you considered the possibility that both are true? If the writer of Romans is the same as the writer of Corinthians (both writings claim to be written by Paul) then, this one writer believed that both propositions were true. As you pointed out, Rom 13 describes national governments to be "an arrangement of God. And he believed that, "the things the nations sacrifice they sacrifice to demons and not to God." (1Co 10:20) So, how could Paul think that both conditions were true?
When Satan tempted Jesus in the wilderness, one of the temptations involved showing Jesus all the kingdoms of the inhabited earth. The devil then tells Jesus, "To you I will give all this authority and their glory, for it has been delivered to me, and I give it to whom I will. (Lu 4:5-6 ESV)
The devil is claiming that authority over the nations was "delivered" (or "handed over," or "relinquished") to him. The passive verb suggests that it was God who did this 'handing over.' (What is known as a "divine passive"; same verb used to describe Jesus being "handed over" for execution.) And Satan knew who Jesus was and where he came from. Logically, there must be some truth in the statement for it to be a temptation for Jesus. Satan is saying that he got the authority from a source that Jesus would view as legitimate, his own Father.
Rev 13:2 describes the devil (the "dragon") as giving the 'beast out of the sea' "power and a throne and great authority." This order, the 'beast' coming into existence first, and then the devil giving it "great authority," this ordering would place the tower of Babel incident as the point in time in which the newly forming human kingdom was "handed over" to Satan, A sort of 'divide and conquer' event. It is only after that that the great empires of the Bible begin to appear.
Of course, that does not explain why God would want such an arrangement. Anyways, you asked for thoughts. . .