Like a previous poster, the unscented products make me feel ill. I also am extremely sensitive to some scents, especially florals. It's hard for me to choose soaps, shampoos, etc., because even some that seem okay in the store make me sick after about an hour of wearing them, so when I find something that works for me, I stick with it. I was assigned to a patient once who was extremely sensitive to scents, even to the soap I used. I ended up switching with another nurse because I couldn't switch up the products I was using to accommodate the patient and minimizing the amount I used didn't help the issue. I was certainly sympathetic, but accommodating the patient by using unscented products would have caused me to be ill.
I think if it were me, I would look for another physician's office - does your physician practice in multiple locations? If you could smell the scent before you even entered the office, that's pretty extreme. If I wanted to stay with my doctor, I would call ahead and explain my situation before coming in for an appointment and ask if they would wash off/refrain from using the lotion while I was there. I think, in general, people try to be accommodating, and they would probably be more receptive to trying to minimize your exposure during your visits rather than a blanket request for a completely scent-free office, especially if you only visit occasionally.