I would try persuading them to donate to both their chosen charity and an EA charity or EA Fund. You could also help them find a charity that's in the same cause as the one they've chosen but does more effective work. The idea is to grow the pie, not make people move donations from one charity to another.
Often times when I talk to people who already donate to charity, they chose the charity due to a strong personal reason and I feel it is not my place to persuade them otherwise. For example, a veteran that donates to a NPO helping other veterans handle PTSD; a Type 1 diabetic that donates to a Type 1 diabetes research NPO; an adult who was an orphan that donates to local children's hospitals.
On the other spectrum, I know people who just aren't interested in charity at all - which I argue is most people in the US.
Perhaps the easiest would be people who are donating to charity, but don't have a strong personal tie to the charity?
Donors without strong personal ties do seem like the most likely "market" for EA messaging, alongside people who want to give but are worried the money will be wasted. I agree that trying to persuade dedicated donors to switch charities is usually a bad idea, unless they express curiosity first.