The noun phrase "government offices" makes more sense than the abstract "levels of government" as examples of publishing institutions. However, there arises the contradiction between "government offices" and "research organizations and learned societies," which may not be governmental.
The inclusion of "while" thus becomes a secondary issue — I think it can be used to express contrast rather than time, being similar to "though." However, I'd suggest leaving it out.
The primary issue is then how to render a reasonable noun phrase that combines state and private institutions. Here's my revised version:
He would send a letter of request to the publishing institutions, including all levels of government offices, mainly ministries and national agencies, as well as private entities(,) such as research organizations and learned societies.
I changed "focusing on" to "mainly" to avoid the coexistence of two present participles, including and focusing, as well as to prevent the misleading adjectival interpretation that "focusing" could have after a noun.