御家来としては国学心懸くべき事なり。
ごけらいとしてはこくがくこころがくべきことなり。
心懸く is the classical form of the verb 心がける. Since the classical auxiliary verb べし is attached to the predicative form (終止形) of a verb, it's read like this.
べし is still used in modern written Japanese, and it's read こころがけるべし, as Majestic-san wrote.
The 3 kanji between 国学 and 事 are neither compound words, nor do they make as standalone any sense to me.
It's the word order as 漢文, i.e., Classical Chinese. Thus, the order is "auxiliary verb --> verb", not "verb --> auxiliary verb (心懸くべき)". 漢文 was used in formal documents, like Latin in Europe. Educated people could read and write these sentences in that era.
Well, since I assume the author is not so conceived to use sonkeigo on himself, it kinda seems like a preface, as it changes the whole meaning what I stated apriori:
These are the analects of the retainers from the Nabeshima clan/way of thinking of the Nabeshima clan.
Your initial interpretation "As a retainer one should put effort into ~" is correct, as Majestic-san wrote. 御家来 is a 丁寧語 to the retainers, not the author himself.
今時国学目落としに相成り候。
いまどきこくがくめおとしにあいなりそうろう。
今時 means "nowadays", not "until nowadays".
目落とし is the noun form of 目落とす, meaning "being ignored/neglected" here.
相成り is the -masu stem(連用形) of 相成る, which is a polite form of なる "to become". Since 候 is attached to 連用形 of a verb, it's read like this.
候 seems to stand for "to be" である、sometimes as a -masu stem?
候 can be attached to nouns, adjectives and verbs, so it's both です and ます in modern Japanese. Note it's about the function, thus, it's not completely the same as です/ます.
In the sense of "to be [at the mercy of]"... I guess it is a retainer spoken style.
Your interpretation is not correct. It's the author's です/ます (respect to the readers).
にである is different form にある. They are completely different.
御先祖様方の御苦労,御慈悲を以, 御長久の事を本落申為二候
御先祖様方の御苦労、御慈悲を以て、御長久の事を本づけ申すために候。
ごせんぞさまがたのごくろう、おじひをもって、ごちょうきゅうのことをもとづけもうすためにそうろう。
本 is usually written as 基 in this meaning.
本落 is not a common wording.
Note that you need to interpret the whole sentence. In other words, it's impossible to understand your quotation correctly without the beginning of the sentence 大意は御家の根元を落着け、, especially the topic of the sentence 大意は. (Furthermore, even this complete sentence doesn't make much sense without the context.)
I agree with Majestic-san. I don't think it's useful for you to become to be able to read such like 国学可心懸事, not only for learning modern Japanese, but also for learning Classical Japanese. It's at least not an effective way of learning. For example, how about the reading of 不可立入芝生? It's hard to read correctly even after knowing 可心懸, isn't it?