The way I understand facism is that it's in the purest sense, extreme ultranationalism. Unlike the other two major ideologies in WW2 (Democracy and Communism) facism was by its very nature not internationalist as usually the goal was to further the goals of the state and maintain the superiority of the race.
However, in the Spanish civil war, for whatever reason facists sent volunteers to other facist regimes. Why would they do this? If in their mind facism is the best form of government and they care for no one but themselves, why would they want to spread it? Wouldn't they want democracy or communism to collapse on itself?
Also, Hitler believed the Germans to be superior to the Italians and Mussolini saying the opposite that the Germans to the north of the border were descendants of barbarians as compared to Italy laying claim to the legacy of Rome. As for the Japanese, Hitler viewed them as simple "culture bearers" who would eventually become an enemy of Germany.
With all these feelings of inherent superiority to each other, how did axis powers manage to cooperate for so long without their pride getting in the way?
While the Axis powers did work together to some extent in many ways they didn't work together, one example of the ways in which the axis powers didn't work together was the fact that they didn't share military technology with each other. For example the Japanese would have to purchase military designs from the Germans in exchange for transporting rare metals to Germany. Whereas it would have been more efficient for the Axis as a whole to share all of their knowledge with each other.
In some ways it would be more accurate to view the Axis as a loose coalition of hegemonies instead of an alliance. Each hegemony was trying to achieve its' own self interest instead of aiming for collective goals. For example the Italian invasion of Greece in 1940, or the Japanese attacking the Americans instead of attacking the Soviets in 1941. (This was partially a result of internal Japanese politics between the army and the navy rather than a grand strategy for Japan)
Also the view that the Axis wished to spread fascism isn't entirely accurate; it would be more accurate to say they wished to spread governments that were allied with them, and would help them to achieve their own aims. One of the ways Spain being fascist benefited the Axis was that Spain wouldn't become communist. A communist Spain would be one more enemy that the Axis would have to fight, and which would trade with their other enemies like the soviet union. However if Spain was fascist instead the Axis could trade with Spain; and having someone to trade with can be very advantageous. Fighting in Spain also gave the Axis somewhere to test out their weapons, and to adapt their fighting doctrine in real world conditions.
I would also like to mention that who we in the modern day view as fascist largely depends on who the Axis believed would be advantageous to ally with. For example it could be argued that the Falangist government in Spain was from an ideological point of view less fascist than the 4th August Regime(also known as the Metaxes Regime) which ruled over Greece during WW2 (whom Italy declared war on due to Italian self-interests).
The axis never really co-operated as; "cooperation" only happened when one side dominated the other. For example when the Grand Council of Fascism in Italy decided to remove Mussolini from power and switch sides to the Allies in 1943, Germany invaded and overthrew it, cooperation between Germany and Italy increased after this because Mussolini required German support to remain in power, and was therefore willing to do what they said.
In conclusion you're completely right in your statement that fascism wasn't internationalist. The axis wanted to selectively spread fascism to gain allies they wanted. While your description of the way the Axis viewed each other is slightly exaggerated (largely due to their views about each other changing from time to time). The Axis managed to cooperate despite their feelings of superiority because it was practical to do so.
Sources:
Fascism in Greece by Harry Cliadakis
Fascist Italy by John Whittam