I've solved this issue, but figured I should write it out in case anyone else comes looking! I wanted to post an article on arxiv that I had written using RMarkdown and the rticles package. Since arxiv wants source files, I had added keep_tex: yes to my yaml header, which left me with the .tex file I needed. But, I also needed a bbl file and couldn't figure out how to get it. Here's the trick!
tinytex::latexmk(filename, clean = FALSE)
Knit your document, then run that code on the resulting .tex to get a bunch of aux files to persist, including the .bbl