I did a bit of reading myself, and it seems as if Loricifera are the only multicellular obligate anaerobes.
I see your point with niches for anaerobic organisms in deep water. Many organisms near the hydro-thermal vents use anaerobic cellular respiration.
From my understanding of Loricifera, it seems as if they would utilize chemosynthesis as their method of obtaining energy.
Loriciferans are found exclusively in marine habitats, and live in the spaces between sand grains or in the mud at the bottom of the deep sea. The nanaloricids prefer sand with low levels of detritus (material derived from the decomposition of once-living organisms) or clean shell gravel, whereas the pliciloricids are often found in such deep-sea sediments as the white abyssal Globigerina ooze and the red deep-sea clay from the hadal zone (below 20,000 ft; 6,100 m).
--From Encyclopedia website: Loricifera (Girdle Wearers).
Lociferans live in a rather oxygen-less environment.
So you're right about niches in the deep sea.
But typically the deep sea has extremely high levels of pressure that few multicellular organisms are built to handle.
Additionally, I was able to find a few other obligate anaerobic Eukaryotes, but those were Protozoa, so thereby not unicellular.
About it fundamentally being difficult to be exclusively anaerobic while multicellular, with multiple cells, there is more upkeep and more energy is required to run the "system". Aerobic processes tend to be a lot more efficient, producing more ATP for usage.
Only 1 ATP (from my understanding) is produced from this, whereas with aerobic respiration:
Aerobic ATP sum
Much more ATP is produced from aerobic processes.
In the end, the majority of obligatorily aerobic organisms are unicellular, or are extremely small.
Their sizes range from 100 µm to ca. 1 mm.
So... that's why multicellular organisms are typically carry out aerobic cellular respiration.
Also, about the fungi you mentioned, it goes through Horizontal gene transfer, which unicellular organisms typically use:
Horizontal gene transfer (HGT) or lateral gene transfer (LGT) is the movement of genetic material between unicellular and/or multicellular organisms other than by the ("vertical") transmission of DNA from parent to offspring. HGT is an important factor in the evolution of many organisms....
Most thinking in genetics has focused upon vertical transfer, but horizontal gene transfer is important, and among single-celled organisms is perhaps the dominant form of genetic transfer.
--From Horizontal Gene Transfer Wikipedia page
So yes, it seems as if the fungi that you mentioned and other similar types of fungi would be strictly unicellular, such as yeast (thought yeast is a facultative anaerobe).