The fansubs always translated the name of the agency the Arashi worked for as the “Occidental Investigation Office,” so I was surprised to see it called the “Office of Barbarian Knowledge Enforcement” in the official translation.
Transliterating what they’re saying, you get “Bansha Aratamesho”. I put this phrase into hiragana, ばんしゃあらためしょ, and googled it. The official page for the series came up, as did the official kanji for the name: 蛮社改所. Obviously, since this group was made up for the show, this term doesn’t appear in dictionaries. Here’s how it breaks down:
蛮: ban, “barbarian”
社: sha, “association, society, etc. (or the counter for those things)”
改: this kanji can be part of a verb meaning “to inspect”, which I’m guessing is the intended meaning here.
所: this is just weird; alone and pronounced sho, it only appears in dictionaries as the counter for places.
Searching the phrase in compounds yields better results. According to this page, 蛮社 is short for 蛮学社中, bangakushachuu, which refers to the Western learning done by samurai attendants. Meanwhile, I found the compound 改所 along with a different starting phrase. 貫目改所, or かんめいあらためしょ, refers to an office under the Edo shogunate that tracked the weight of shipments moving along highways, according to this page. Since 貫目 means “weight”, you can infer the meaning of 改所 to be something like “inspection office”.
Based on this, I’d say 蛮社改所 could be translated as “Western Learning Oversight Office.” I can see where the translators got “Office of Barbarian Knowledge Enforcement,” though, given that there’s no distinction between “barbarian” and “Western learning” in this time period.
(Fun side note: Here is an article about 蛮社の獄, or “Jailed for Western Learning,” that mentions Takano Chouei, who happens to be one of the many historical figures featured in Ayakashi Ayashi.)