That's why it's a challenge.The problem is: in exile where to?
Assuming a WWII like OTL, nobody will want them out fear of enraging the Allies.
Can't be done. No, really, it can't. It's one thing to host a number of officials from an Axis nation, as Argentina and Spain had done (and for which they received a lot of flak for and proved extremely detrimental to their global relations/trade/etc.). Its another entirely to host a government in exile. No one, not even Franco or Peron, would be so stupid as to attempt it, and all the economic and political power would be brought to bear so as to force said nation to eject said government in exile. Even Italy would be a major stretch despite the lack of..............."major issues" that Germany and Japan had during that time.
1) have a Great Power do it (most likely the Western ones, out of anti-Communism; a very small chance for Italy, i'd say zero for Germany and Japan).
2) have the Exiles setting up shop somewhere against the will of the local goverment, possibly in secret.
Agreed. Italy is the only one that could possibly do this, but considering the conditions that even the Italians were forced to meet following the surrender of Germany, I doubt it is possible even with a Communist mainland; a Mafia regime has a stronger possibility then a Fascist one (and no, while they may be both authoritarian, they are quite different, and neither would want anything to do with the other).
That isn't a government in exile though. A government in exile has to be public in some shape or form.
Ante Pavelic formed an exile government OTL
The Vichy French are able to work with the Japanese in Indochina, which remains in Vichy hands throughout the war.
The area remains in Vichy hands for a few years, it doesn't last long.
Where? For how long?
In Argentina and IIRC it transformed itself into a partywhen Croatia became independent in 1991-2
Fascinating. Was it recognized by anybody at all? Did Argentina and Tito's Jugoslavia had diplomatic relationship ? Sounds like a pretty big stumbling block to any sort of constructive interaction.
EDIT: considering that Pavelic guys were the ones who horrified the Nazis for their cruelty...
I could also see, if Puyi got to Japan, and managed to stay out of jail to see the CIA under Allen Dulles trying to stand him up as some kind of counter to Red China, its just the dumb kind of plan they loved in those days