Unless your dog burns fossil fuels you most likely only need to pay carbon tax on the dog products you use that were made using fossil fuels. Dogs don't produce a lot of Methane either, because they're not ruminants.
Cattle only burn fossil fuels to the extent that transporting water and feed, fertilizing that feed, and manufacturing and transporting fencing materials, antibiotics, etc, use.
Guess what? Your dog has roughly the same impact making and transporting his food, his water, his veterinarian supplies, any bedding you've purchased, bags to dispose of his feces, collars, leashes, toys, etc.
I don't have hard numbers but intuitively I'd guess dogs and cattle have approximately the same climate impact on a pound-of-bodyweight basis.
Dogs probably have slightly more, since they live in much more luxurious digs than CAFO cattle, eat higher quality feed, get to ride in cars, etc.
I would be fine with paying a carbon tax on my cat, the meat he eats (all chicken, at least, and not that much) has some footprint (pawprint?) after all. I do pay for some extra offsets via Wren, so maybe he's covered.