Birchwood Casey Brass Black contains acids, but some of the ingredients are used in cold blues. Here's an MSDS listing the ingredients:
https://birchwoodcasey.com/files/datasheets/15225%252c-15232-Brass-Black-Metal-Finish.pdf
Hard to say what it might do to a fresh rust blue job without actually trying it. I used some years ago to darken the shiny brass furniture on a flintlock hunting rifle, and it was pretty durable and held up well, at least until natural tarnish took over. I think Sharpie permanent marker would come off when using cleaners and gun oils fairly quickly, but it could always be re-applied. I'd be inclined to try using the brass black in a very thin line before applying your rusting solutions, and then rust and boil normally taking care not to card aggressively on the braze joint that has already been darkened.
