I have been knocked down by other people's perfume for about 30 years now (severe headache, nausea, dizziness and confusion, inability to function or sleep for up to 4 days). It really takes a bite out of any possible social life, except for a weekly discussion group that has voluntarily stopped using scent (for the benefit of 3 participants; at least I'm not the lonely weirdo in the group).
My mother uses a strongly-scented shampoo that I've begged her not to use just before I have to drive her somewhere, but she does it every time… then complains about my having to have the car windows open. I tried explaining it to her several years ago, and she wept with fury and personal insult.
So, I may not be the person to give useful advice. The nearest I've come to a successful communication is, "I'm so sorry, but I'm having a very difficult reaction to your perfume. I must sit farther away from you." On only ONE occasion, this has resulted in a caring friend actually choosing to stop using the scent so we could stay close. A couple of those I've told have either continued using the scent, sometimes giving me an "up yours" look when they can see it hit me, or become angry and pointedly avoided me.
It's almost impossible to bring it up to strangers. On the few occasions where I've tried, people have either failed to comprehend that perfume could cause anyone misery, or become quite angry at my "rudeness." I never say anything insulting, just "Excuse me, I'm having a bad reaction to your scent, so I must move away from you." On one occasion, the "friend" I had been talking to kept moving in closer, arguing that I was imagining illness. I ended up vomiting in the bathroom 10 minutes later. I couldn't even tell her, since I knew by then she wouldn't believe it was her perfume.
I made a small card that I tried handing out for a few months: "The person who gave you this card has had to leave because strong scents make her ill. This is a common problem, with at least 1 in 20 people having uncomfortable or serious reactions. Kindly consider their need to breathe safely before refreshing your scent again." I handed out the card, then scooted out of harm's way. So, I never got positive or negative feedback, and finally got bored with it. I doubt that the recipients were receptive.
I wish you success. If you find a way to communicate that's non-offensive and effective, I do hope you will share it.