At this level, things often get a bit more flexible than necessarily applying for a particular program. Especially beyond residency and into fellowship, if you really want a PhD, it's often more a matter of finding an advisor who wants to take you on and fund you and getting the okay of your program. If you're in a program with protected research time, there is probably a way to try to make some graduate training part of that time. If the fellowship/residency director isn't into it, then you will have a hard time trying to make it work.
For example, for the Stanford ARTS you mentioned, you apply for that program once you are in residency. And that's not the only way to get a PhD at Stanford as a clinician. ARTS is about funding, if you can find and advisor who is willing to fund your research/graduate training and salary, then you can go that route. However, there are also other ways to fund your graduate training, including some NIH training grants, etc. Your application to a PhD program is often about money, and if you come in as a credible person in that area with publications and with your own funding and an MD, then you're usually in a good position.
There are all sorts of possibilities too, I know some people who in their protected research fellowship time somehow set up admissions to graduate programs in Europe, which are typically shorter requirements (France, Germany, England, etc.) than the US, and got PhD's (or D Sci or whatever) there by extending their fellowship time only a little bit.