First off, I agree that your friend is reacting strongly because she knows you’re right about church teaching, and she feels guilty about not following it. But you can’t really go right out and say that.
Second, you should realize that a lot of older Catholics were lied to, by either priests or by other trusted figures. There was a lot of assumption, before Pope Paul VI came out with Humanae Vitae, that of course the Church was going to go along with all the other churches, and that of course birth control was going to become magically church-approved. When Pope Paul VI asked for a panel to study this stuff, it was leaked that a lot of the bishops on the panel were totally okay with birth control. Priests were telling American couples that they could just go ahead and use the Pill as much as they wanted.
And then Humanae Vitae came out, where Pope Paul VI strongly reiterated the Church’s teachings, and where he pointed out that birth control was disrespectful to the whole idea of married life or even of sex. People freaked out. Some of them freaked because their priests had taught them wrong, but some of them decided that the Pope must be wrong. And a lot of priests were defiant and continued to teach people that birth control was okay, for all kinds of reasons.
The other thing to realize is that the old version of NFP, what they called “the rhythm method,” was not terribly accurate because it was new. It was also constantly ridiculed by the media, which was not really fair. (Nobody ridiculed the failure rate of media-approved forms of birth control.)
The very idea of refraining from sex at any time, for any reason, was supposed to be something that would instantly destroy a marriage. A wife who wouldn’t have sex with her husband, even for health reasons, was cold and hateful. A man who wouldn’t have sex with his wife, even for health reasons, was ungenerous. Any lack of sex, and any lack of fulfilling a spouse’s strangest fantasies, was just driving the spouse into the arms of another. And so on, and so forth. The Sixties and Seventies were a weird time.
A lot of these ideas persist, even among younger Catholics, because they hear this stuff from their families or from their doctors. They may not be fully responsible for their sins, because they have been seriously misled.
So yes, you should be gentle. But you should not back down. The teachings of the Church on this topic have been constant since the first century, in the Didache and the Bible.
But you can expect people to be angry, whenever you cover a topic they’ve been misled about. They will be angry and disbelieving at you, first. But if they recognize the truth, they will probably end up angry at whomever misled them. Sometimes it takes a while to get over all the anger. I’ve been ragingly angry myself about being misled on much less personal topics than contraception and marriage, so I imagine your friend is pretty mad.