I think Germany should press the claim regardless. The only reason they didn't claim south tirol from italy was that Hitler and Mussolini were Dictator buddies, both facist and more or less equal partners. Also Italians were not really categorized as subhumans, unlike every other force in the area except maybe Hungary and Romania(all others were slavs in some way). And you can be sure as hell that the german leadership would NOT have accepted the territory in Hungaries or Romanias hands. The problem with post WW1 borders was that they were largely drawn based on historical borders centuries old and not quite up to the task (Save for the polish territories excluded from germany by popular vote of the inhabitants).
The borders of Czecheslovakia were those of historical Bohemia-Moravia, even though the language border had since shifted in the centuries before, so quite some Germans who had lived there for generations (or Germanized Czechs who had lived there even longer but considered themselves German since generations) suddenly found themselves inside foreign borders. This was a problem that the victors of WW1 were aware of, of course, but they did it anyways to spite germany, the loser, and to strengthen their new allies.
Just think how much sorrow and misery could have been avoided had someone just redrawn the maps based on Cultural belonging instead of stupid historical claims long overtaken by reality. Don't get me wrong, Germany would have started something anyways sooner or later, Germanys real problem was the high number of desilusioned, desperate veterans, highly armed and ready to use violence, and the explosive nature of the society at the time. Just read up on the "lost generation" to know why killing Hitler won't necissarily prevent it all, heck you could have made it worse because competent, not insane people could have tryed what he tryed. But such redrawing would have robbed Hitler of a lot of Momentum and support, and after WW2 would have prevented the massive displacement of Polish people by the SU into the now polish regions where they displaced the germans, had they kept those borders. One can dream. Anyways.
So the reason that Germany claims those regions is to get popular support from the populace as the protector of all germans. Unlike south tirol, Austria and the Sudetenland and Danzig (70% German at the time and annexed by Poland even though it was supposed to be independant) could not be ignored because they were too large, too populus and in the hands of the ENEMY. I can not come up with any scenario in which Germany would accept these Territories in anybodies hands but their own. If Italy takes it, the area will become too large and too historically unjustifiable to let them keep it. If anybody else in Europe takes it, Germany WILL demand it as well. If Austria takes it, they will be Anschlussed anyways.
I don't want to get too deep into the inner political imperatives that mandates them to take it, Lost Generation, Nationalism and the Unification of all German peoples, Protector of Germans yadda yadda.
So yes, Germany should allways try to take it and start hating you for having it if you are not austria