Beyond what you're saying (which is quite true), if you're rescuing dogs, as I mostly do, the only way you can say there is a net increase in carbon is because I'm providing a better quality of life for my dogs (which undoubtably increases carbon emissions) and causing my dog to live longer than it would if it was wild or killed in a shelter.

I don't particularly see "reduce carbon impact by killing dogs/animals" as being a winning slogan.