So there's definitely an element of woo-woo to some Functional Medicine docs. This particular clinic near me does use homeopathy sometimes. So far as I'm concerned, that's just giving someone a water placebo. Now there approach is open to using a number of different modalities besides just homeopathy--they're big on diet, supplements, mind-body connection, using meditation, osteopathic manipulative techniques, massage, accupuncture and herbs. And those things seem to be, as far as I can tell, par for the course for most Functional Med doctors. The other thing, being "wholistic" docs (which is sort of a big grab bag term for whatever you want it to mean I suppose), they tend to focus a lot of patient relations, listening to patients, finding more time for patients, etc. And, at least for this doc who talked at our school, his patient ratings on all the online websites are ridiculous. 5 stars all around. Patients love him. That's going to be a function of the physician more than the approach. That said, I think the people who embrace functional med are going to be the physicians that want that connection with their patient.
I think that, on average, if you're looking for someone open to your particular diet and health ideas, functional medicine doctors will be more receptive. They might have some other crazy dietary ideas of their own, but they're less likely to freak out that you're eating a bunch of fat and need to be put on Lipitor stat. At least this is the impression I got. The flip-side, despite the Institute of Functional Medicine's website, is that you might get someone who is a little heavy on the woo, and less on the science. Or at least they might be swayed by bad science that fits their worldview over good science. This is the case for most physicians though (and most scientists, and most people in general).
It is probably worth the drive to at least have one appointment with the doctor in question and feel them out. Really the most important thing for you is your relationship with your physician--they could have a similar philosophy on paper, but if you don't like them or trust them in person, they're worthless to you. And especially if you're looking for a primary care doc, you probably want to make sure you like them and trust them. So make the drive and see what they're about, that's really the only way to tell. I like the Functional Med philosophy, they talk a good talk, but the proof is in the pudding, or the stethoscope. I don't know if that answered anything or not, but good luck.