Obviously a true fix should be made for a bug, but in case it will be useful in the meantime I’ll mention that a workaround might be to remove double quotes from the Markdown typographer quotation marks site setting, or disable the Enable markdown typographer site setting.
I am assuming that it is this “Markdown typographer” feature that is causing the replacement of the correct straight double quote character in the BBCode tag with the incorrect “smart” or “curly quote” character.