No, the Bf109 never operated off of a CV....
Well if we are going that route, then we should say, the rooks can't have CVs, because the Germans didn't ever complete any. Oh, and the bish always capitulate halfway through the war and hang their leader on a rope (like Mussolini) and thus you can't fly italian planes once some bases get taken.
the 109T never was in production because there never were any CVs. That being said, the american rides dominate the choices from a CV in AH as we don't have many of the Japanese planes (compared to German/American/British), so a 109 would be a nice temporary stop gap until HTC is able to fill out the Japanese plane set more completely.
FYI, we have German planes that were produced in numbers in the low hundreds, yet we are missing Japanese planes that had production numbers well into the thousands.