In The Art of Fermentation, Sandor Katz notes that he has fermented shiitake and some other mushrooms, but does not specify fresh or dried.
Christopher and Kristen Shockey, authors, owners of Mellonia Farm and the website Ferment Works, include a recipe for Pickled Shiitake in their book Fermented Vegetables that calls for dried shiitake. In the recipe notes they also suggest adding dried mushroom slices to sauerkraut.
Fish is often added to vegetables for kimchi, depending on the style of kimchi being made. Maangchi, cookbook author and blogger, has several recipes for kimchi, like Easy Kimchi with fish sauce and fresh squid and Traditional Napa Cabbage Kimchi with fish sauce and salted, fermented shrimp.
I have not seen recipes for oil-packed fish, but fresh, salt-packed, or dried anchovies would all be worth experimenting with.
I have heard of oil being used in certain applications in lacto-fermented vegetables, but I am not sure how an oil-packed ingredient would interact with the rest of the ferment. For example, I have read blog posts of people use a layer of oil on the top of their fermentation vessel as a type of airlock--the CO2 passes through the olive oil but oxygen cannot penetrate it. I have not used this method, though.
There are also traditional Indian pickles made in mustard oil in sunlight (whereas most lacto-ferments are kept out of direct light or even in darkness).
There have been cases of botulism forming in olive oils flavored with fresh garlic and herbs; botulism bacteria are introduced to the oil from the fresh produce and then thrive in the anaerobic environment of the oil. Acidification is required to treat the produce to prevent botulism. Lacto-ferments create an acidic environment unfriendly to botulism, so the situations are not identical, but caution should be used when introducing oil to lacto-ferments.