Without knowing the type of photographs you're talking about, I'd have thought they wouldn't have too much appeal beyond the dog's owners themselves. On that basis you're better off trying to pacify this owner rather than wind them up further. I suspect if I were in your position I'd craft a reply to her saying something along the lines of, I'm sorry she isn't happy about the photograph of her dog being made available to other people, as result I'd offer her a free print and offer to donate the 'cost' to the charity yourself.
I wouldn't apologise for taking the photo, not asking her permission, or for advertising the print for sale, at the same time I'd also avoid pointing out that you don't actually need her consent - which will probably just aggravate the situation further. Finally, I'd remove the picture of her pooch from the website so she can't then come back a second time moaning that it's still on offer! In practice you probably aren't limiting your sales too much by removing 'her' pet pooch, and for the cost of a complimentary print you'll hopefully have a content pet owner who won't take the matter further. You're unlikely to convince her that you didn't need her permission in the first place, so there's not much point trying!
I have a weekend cottage in a small picturesque Arkansas town. There was a movie being filmed on the street my house is on. The producer insisted on getting a release from me, even though they were not actually going to use my house, rather it "might" be seen in the distance if they shot down the street. I found it odd that they needed a release for this, since all that would be seen was what was publically visible.
First, very good of you to donate the profits to charity.
Second, it's hard for me to think that someone would want a "portrait" of someone else's dog. I can understand, perhaps, a photo that depicts the breed doing what the breed is designed to do, but not an individually recognizable dog.
Regardless, I can understand the owner's reaction. They come in for a sitting (I'm inferring a bit from your original post) and you sell a portrait of their dog to someone else. It's not quite the same as displaying photos as part of your portfolio or capturing their pet's image as part of a scene (say as part of a group of dogs).
But you might find that if people have the opportunity to "opt in" they may be quite willing. A statement to the effect that these charity efforts are supported by sales of dog portraits to owners and others, with the ability to give permission to do so, will garner a fair number of approvals. Adding that a portion of your profits will be donated to the dog charity (following through of course) will probably improve the approval rate. As well as possible sales.