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For example, what case would 'sword' be in the following sentence:
He used the broom as a sword
Is it simply accusative/dative? I don't think this is just synonymous to 'general' in 'He acted like he was a general'. Could it perhaps be instrumental? I'm not sure, though, because it seems to be the broom relating to the sword, while in instrumental it usually appears that the word in the instrumental case is relating back to the accusative.
“As” has a lot of the preposition about it. In your example, it means “in the manner of (a B)” As the “of” is a preposition, the substantive that follows is not in the nominative.
If we imagine the producer of a play who, at rehearsals, has been told that Rita Chevrolet, his leading lady, is sick and he needs a replacement, he might say “OK, use me as her.” He would not say, “Use me as she.”
From this, we can conclude that “a B” would be in the accusative or dative case (probably dative, as "of" took the dative in Old English. https://oldenglish.fandom.com/wiki/Reference_7:_Prepositions)
The instrumental case no longer exists as such in Modern English - its meaning is conveyed by making the noun the object of a preposition (usually "by" or "with")
In Old English instrumentals can be recognized as nouns in the dative case that are not indirect objects and are not preceded by a preposition.
Alfred killed a Viking with a sword.
"Sword" is in the instrumental case because it is the instrument Alfred used to kill the Viking.
However, it is just as easy to think of "sword" being the object of "with," and thus in the dative.
Old English gives a writer the option of leaving out "with" and simply saying "Alfred killed a Viking sword(+dative ending)." (<== note that "sword" is in the dative / instrumental case) https://people.umass.edu/sharris/in/gram/GrammarBook/GramCases.html
as... relates to a previously mentioned person/thing. The case of B depends on the case of the mentioned person/thing it relates to.
He [nominative] acts as a B [nominative].
He presents his [genitive] own position as B's [genitive]. (as... relates to 'he' in 'his' not to 'position'.)
He sees her [accusative] as a B [accusative].
He pays tribute to her [dative] as to B [dative].
Setting aside the issue of -'(s)-genitive constructions, which are not relevant to your question, English common nouns such as sword don't inflect for case. The only type of word that shows case inflection in English is pronouns: e.g. me, him, her, us, them are distinguished from I, he, she, we, they.
The historical distinction between accusative and dative case forms in English has been completely lost, so "accusative" and "dative" can no longer can be distinguished by form even for pronouns. Although the surviving forms are mostly descended from the Old English dative forms, and the surviving functions are descended from both the functions of the Old English dative case and the functions of the Old English accusative case, it is fairly common in the English grammatical tradition to use the term "accusative" by itself to refer to the surviving forms. This is I think why RegDwigнt♦ expressed confusion and disagreement in response to your wording "accusative/dative". My guess is that you are using "accusative/dative" not to refer to two separate alternatives, but as a two-part name for the case taken by pronoun forms such as me, him, her, etc.
Treating "instrumental" case as a third distinct category is another relic classification that is usually not made when discussing the grammar of modern English.
A general rule of English morphosyntax is that the object of a preposition can only take one morphological case form, the "accusative" form. (Setting aside the issue of coordinations involving pronouns, which show variable usage that has been explained in various ways.)
As mentioned in Greybeard's answer and in the comments, the word as can be argued to function in many contexts in modern English as a preposition. Some "prescriptivists" dislike this use of as or say that it is "incorrect", but the syntactic analyses that linguists tend to focus on are about explaining what the body of evidence from speakers of a particular language actually shows about usage ("descriptivism") rather than explaining what certain speakers say usage ought to be.
If we analyze as as a preposition, then according to this rule, a sword must be in a syntactic position that grammatically requires the use of the accusative morphological case-form in English (even though this morphological distinction is not actually marked on a sword).
To confirm that the analysis of as as a preposition is correct for sentences like your particular example sentence, you would have to construct a sentence like this where a pronoun with visible morphological case distinction comes after the word as. It seems difficult to do this naturally. Maybe something along the lines of "While I used boxes as medicine containers, she used bottles as them" could serve. I find this sentence highly unnaturally worded, but it seems notably more acceptable to me than *"While I used boxes as medicine containers, she used bottles as they."
If by "case" you are not referring to anything expressed in the morphology of English, but to a purely syntactic phenomenon, then you ought to try to analyze the construction from the point of view of a modern theory of syntax. I don't know how to do that, so I can't offer anything as an answer in that case.
The prince took on the the peasant girl as his manservant. Should I say,
He took her on as a him.
*He took her on as a he.
Clearly, the latter won't do. So B is objective.
