I saw a pretty smart way in the below YouTube video to get Germany to let you join the Axis as a non-aligned Austria-Hungary.
One of the possible outcomes of the "Claim Transylvania" focus is where Romania refuses to give you land, and you can approach Germany to mediate. If you started improving relations with Germany 45 days prior to this event, and get a favorable RNG result, both Austria-Hungary and Germany will declare war on Romania together in Feb 1938. Once that happens, Germany will let you join the Axis, and Germany will be blocked from the focuses that let them demand Sudetenland.
Unfortunately, given the French guarantee on Romania in the current version of the game, this will start WWII 18 months early. But, given that all the majors are less prepared, it gives Austria-Hungary a good chance of beating the Allies before Barbarossa. You can then switch sides to the Comintern after a while, and then grab the lands you need from the Axis in the resulting peace deal.
The part of the video that shows the Romania early war RNG event is at 18:20: