No, it is correct without a comma. If the visually displeasing nature of
When rebounding becomes important,—as is the case when playing the Lakers—we need to have John in the game.
isn’t good enough a reason for you, a dash functionally takes the place of a comma (or colon/semicolon/parentheses depending on context). You cannot have both a comma and a dash directly next to each other, as is the case for all other forms of sentence punctuation barring parentheses. Think of it this way: If you were to set apart the parenthetical with commas, which would be just as correct syntactically, would you include a double comma? In other words, would you write this?
When rebounding becomes important,, as is the case when playing the Lakers, we need to have John in the game.
Certainly not! You would just use a single comma performing both functions. So why would the rule be any different for a dash? In many cases punctuation marks are subsumed by others. Take dialogue as an example; when a quotation ends with a question mark, that mark also ends the whole sentence, as in
John asked, “Where are you going?” (Notice the lack of a period after the quote.)
You say the subordinate clause “is a separate matter from the tangential phrase between dashes,” but I disagree. The parenthetical is directly related to the subordinate clause; it describes a certain instance of when rebounding is important. If you were to add a comma somewhere, it would be best placed after the second dash. But once again, it is best left out.