Your research interests are very specific. It’s unclear to me if you’re willing to visit a bunch of palace and cathedral treasuries just in the hope there may be some religious vestments of interest on permanent display. Or do you hope for specialized archives that might require an appointment?
There might be some fabric fragments in the splendid old synagogue museum in Erfurt. Did you know Erfurt made its fortune in woad dye? I also suggest the schloss in Gotha, especially since it has one of the three major surviving court theaters of Europe. I wonder if the UNESCO WHS Dessau-Worlitz park has any textiles in the many buildings?
Although it's not that easy to get to from Erfurt, I should mention Quedlinburg. It's mainly visited for architecture, but it has a small cathedral treasury that has some of the most important individual objects of that kind in Germany, which is saying something. It was plundered in WW II, and later returned. It's mainly things like book bindings made from classical artifacts and jewelry, not so much fabrics. I do think that the former East Germany is a good place to look, because there are many places that time can still be said to have passed them by. Don't put off Prague if you haven't researched it yet, or the heavily touristed but worthwhile Cesky Krumlov,.
I wonder if you should select some medium sized historic German cities, like Aschaffenberg, and see if there are collections you want to see there. I mention Aschaffenberg because I've been there, and both the huge Schloss and its' associated Pompeianum should have some objects that involve textiles or garments.
It's a more obscure thought, but I've been to some tiny towns, like Lörsch, where the (modern bronze) replacement for the town fountain displays the home-needlework trades that built the town. But such towns (despite in this case, having a UNESCO WHS mostly-vanished cloister!) are too small to support the scholarship and expense of a real museum. Now, that cloister has a swell museum, but it tends towards early 20th century objects, other than the tobacco industry part of the town's history.
Edit: I see that Leipzig has a substantial Museum of Applied Arts, which will have textiles and maybe garments. Have been to the excellent Fine Art museum there, but not the M.A.K. The Bavarian National Museum in Munich is Decorative Arts, but I only remember a whole case (?) of John The Baptist's head-platters!