I’m not sure I buy that it’s just because of a ‘religious’ reason. Like the other user pointed out, the Emperor Meiji broke with traditional Shinto practice and did publicly eat meat to symbolically break with tradition. Much of the State Shinto policy attempted to break with more traditional Shinto (and especially Buddhism) . It seems more practical than theology.
At the absolute best, Emperor Meiji might have known it was a deficiency of some kind in his mid 30's. Even then it's unlikely as it didn't become internationally known until at least his late 40's.
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The vitamin deficiency you're talking about is specifically beriberi. A B1 deficiency.
For any of this to make sense you first need to understand that white rice is made by removing husk, bran, and germ. This had two advantages to ‘regular’ rice. It stores much longer and it’s easier to digest. The downsides? Well, it’s a pain to remove all that and polish the rice. It’s only feasible to eat it over brown rice if you’ve got the time or money/power to have someone else do it. At least until rice polishing machines were invented. Also, that stuff on the outside? Turns out it’s pretty important for living because it has vitamin B1. You can get some B1 from some fish, but you would have to balance it out. For a small, resource poor nation like Japan a domestic livestock industry wasn’t practical. The only way then is importing animal meat.
But this is all with hindsight. As far back the 6th century beriberi was already known in ancient sources across China. But it was thought to be related to a weakness of the feet because of the numbness and difficulty walking are major symptoms. Like the miasma theory in the West (hence mal aria, "bad air"). It wasn’t clear what exactly caused these diseases.
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Beriberi was actually somewhat common among the Japanese upper class well before the Meiji Restoration. Meiji’s uncle and the next to last shogun of the era, Tokugawa Iemochi and his wife Kazu-no-Miya were probably a result of this. Iemochi was a devout practitioner of Buddhist asceticism. Particularly, that it was most noble to live a simple life with little luxury. Guess what food fits those criteria? After his death of heart failure at 20, Kazu-no-Miya would become a Buddhist nun and die of the same disease as her husband 11 years later (and help to negotiate a peaceful end of the Boshin War).
By the 1870’s, rice polishing machines are invented, and for the growing Japanese military this seems ideal. The rise in beriberi disease starts to become a noticeable concern around 1884 (When Emperor Meiji is about 33) particularly by Takaki Kanehiro a doctor in the Imperial navy. He notices that officers are contracting beriberi far less than the poorer crewman. White rice was provided by the Navy, but anything else like meat or fish was extra that the officer had (and often did) to provide themselves. After testing with two crews, on the same route and only diet changed, the cases drop from about 45% to 4%. He theorizes it's a result of a protein deficiency. As a result, the Navy starts feeding their crews a more varied diet with meats, beans, etc.
The Imperial Japanese Army however, isn't quite as impressed. The Army’s doctors believe it’s an infectious disease. After all, this is when Louis Pasteur and Robert Koch are showing the world the germ theory of disease. Hillary A. Smith argues that there was underlying assumption that the European model of medicine is inherently more sophisticated and that a dietary explanation came across as 'primitive' and holistic in comparison. Regardless, Takaki isn’t fully able to convince the Army until after both the Sino and Russo Wars show the Navy losing far less soldiers to beriberi than the Army. This is around 1905 (Emperor Meiji is 53 and 6 years away from death).
In 1897 Christiaan Eijkman’s work identifies something essential in brown rice. A vital amine or 'vitamin' (technically not all vitamins have an amino compound which encouraged shortening/renaming it to 'vitamin'). Though it will still take a few years for others to isolate and identify B1 proper. Suzuki Umetaro in 1910 and Casimir Funk in 1912.
Eijkman received the Nobel Prize for Medicine in 1929 alongside Sir Frederick Hopkins. A small peninsula in Antarctica is named after Takaki Kanehiro thirty years after, in 1959.
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Sources:
Oberländer, Christian. The Rise of Western “Scientific Medicine” in Japan: Bacteriology and Beriberi. From: Low Morris. (ed) Building a Modern Japan
Smith, Hillary. Forgotten Disease: Illnesses Transformed in Chinese Medicine
Otsubo, Sumiko, Review of Beriberi in Modern Japan: The Making of a National Disease by Alexander R. Bay, Isis - Journal of the History of Science Society, 2014
Jensen, Marius. The Making of Modern Japan
Hanley, Susan. Everyday Things in Premodern Japan: The Hidden Legacy of Material Culture