I think you are right that a closer look at the makeup of the 33-member Duquesne Nazi Spy Ring might be instructive here.
While every member of the ring had pre-existing ties to foreign countries*, not one of them came to the USA as a refugee from Germany. Several in fact didn't come from Germany at all. The ringleader was South African.
There were in fact several people of Jewish extraction who were convicted of spying on the USA during this period. However, what they tended to have in common was that they were not themselves refugees but rather descendants of refugees, and they spied for the Soviets (their enemies), not for the Nazis.
There was at least one Nazi spy caught in England posing as a refugee from an occupied country, Willem Ter Braak. Presumably several more spies caught early on in their (often comically bad) infiltration attempts were planning on doing the same thing. Of particular interest would be Jose Walberg, Karl Meier, and two other Dutchmen (whose names I couldn't dig up), who again were comically incompetent, but had forged Dutch refugee papers on them. This spooked the British enough that they detained and started screening all Dutch refugees.
There was another set of refugee Nazi spies in England who, as per the pattern in the USA, were not refugees from Germany, but rather from the USSR (their enemies). Presumably they acquired their love of Nazism from their hatred of the Bolsheviks. Examples of this class are Marina Lee and Vera Von Schalburg
* - The exception here is one Kansas native who was the ringleaders girlfriend, and was convicted basically of not reporting on him.
I think if one is to take a lesson from all this, its that refugees themselves don't tend to be a danger, outside of spying for the enemies of the regime they fled. However, the existence of a refugee community does provide a tempting bit of cover for those who might try to sneak in through other means. The really big danger (such as it is) seems to be in your second-generation citizens.