I think, the handwritten font, that is used by the plotter, does not support german umlauts. But if you create your own handwriting font, this might be a fun idea to try to get away with.
Its much more difficult than that to be actually believable. As u/Luftruessel said, theres a great video from “Stuff Made Here” where he goes deep inside the topic and tries to fool a graphologist.
As part of the copy chain, you need to feed the ChatGPT output into a handwriting neural network you trained in your own handwriting, then have the 3D printer draw it.