I wrote the following sentence in an essay: "My idea was that, since A is B, B might be A". However, word count matters in this essay, so I'd like to know if the "that" is optional in this sentence. More specifically, I want to know if "My idea was, since A is B, B might be A." is also correct.
My idea was that [since A is B, B might be A.]
Yes, the subordinator "that" is optional here, where it introduces the bracketed declarative content clause.
But note that in such clauses, "that" is sometimes inadmissible, for example when it is complement to a preposition like "before", as in in *"I left before that he arrived".
And sometimes it is obligatory, for example when the content clause is subject of the matrix clause, as in "That I need help is clear". Elsewhere it is generally optional, though it is more likely to be omitted in informal than formal style, and after short and common verbs.