Ok, I'm finally getting back to this thread now that my schoolwork has slackened a bit. Thanks to everybody for the informed responses.
Moving back to the original topic, I guess it isn't really viable for the Germans to license an IJN carrier, and McPherson and Calbear, thanks for the info on the Fi-167 and the BF-109T. I researched it some more and it looks like you guys are right, I underestimated the effect the additional weight would have on the T and the unsuitability of the Fi-167.
Thinking about the Aquila, this is my idea. The Italians were kicking around the idea of building a carrier for all of the 1930s, but it never got done. My POD is that the Spanish Civil War for some reason doesn't happen (let's say the worker's militia are able to suppress the Nationalists in Seville and the whole coup sputters out). The amount of money that both Italy and Germany spent on that war was INSANE; Italy dropped $415,000,0000 in 1939 dollars on it (that amount of money would have been enough to build about ten King George V class battleships for comparison). With those increased resources, Mussolini gives them the go ahead in late 1936. It's plausible, the countries that operated them at that point were mostly countries like Britain, France, the U.S., and Japan that had far-reaching colonial possessions, and Italy was in that club after they conquered Ethiopia. With more available resources, Mussolini might well have decided to do it. This triggers a mini-arms race, with the British authorizing another Illustrious-class, and Germany responds to the trends and decides it would be good to have at least one proper fleet carrier. Nazi naval strategy seems like it was literally random, so who knows, they probably could do that. So they modify the Graf Zeppelin design to reflect this...no more excessive armor and guns, larger air wing. That's the ships.
The second part is the planes. The Bf-109T wasn't a great design, but the carrier-borne version of the Reggiane 2001 had promise. The Germans reviewed and rated it more highly than their own. It could also carry torpedos. Both were powered by the DB-601 engine. My idea was, what if the DB-605 was fielded earlier instead of in 1942 as IOTL? That was the more powerful evolution of the DB-601, and it was basically the same as its predecessor that dated from 1935, except that the German engineers figured out they could drill the cylinders four centimeters wider and get a huge performance boost. The Bf-109 and Reggiane 2005 (basically just the 2001 with the 605 instead of the 601) both maxed out at over 400 mph with it. If the Daimler-Benz engineers drill the cylinders a little wider to create the 605 in like 1937, it would allow them to create the Bf-109G and the Reggiane 205 early, which would give them a carrier fighter with extra horsepower to handle the additional weight of carrier fighter equipment. For a bonus, if the Italian design turns out way better, the Nazis could order it to get a better fighter that can also fill the torpedo bomber role. There was precedent...the Luftwaffe used foreign-sourced planes, the Czech Avia B-543, in their carrier program.
I think this is a pretty realistic POD involving earlier decisions that were made later and minor modifications to engine technology. What do you think? I am trying to troubleshoot a TL idea that I have, I'll admit that, and I'm trying to do my own research, I was just wondering if anyone here had any thoughts because there's a lot who know more than me. If it just isn't plausible, I can accept that.