Sit on top kayaks are typically the MOST stable. The ones designed for fishing are awesome in that regard. You also don't have the issues with getting in and out of them if you fall over but seriously I have been in really tall swells from motorboats in ours with zero trouble. SOTS are also self bailing so they can't sink or fill with water. Fishing kayaks are designed with enough stability you can stand and fish.
We have Wilderness Systems Ride 135 fishing kayaks and it is great with a lot of back support (we have the seat)..The problem is finding a good kayak that the dog can be comfortable in. My husband had spinal fusion this fall and for flatwater it is not hard on him. (Other than transporting it-we made a trailer and have a kayak cart and that helps a lot)
Perception just came ut with one I would like to try. It is a cross between a SUP and a SOT that looks promising for use with a dog.
https://www.perceptionkayaks.com/us/products/hi-life-110
If you have a kayak shop they all often have "demo days" where they take the kayaks to lakes and let you try out various models. Also many sell of their rentals at the end of the summer for about half price.
The old town predator MX is another one I would look at because it does not have a console between the legs like a lot of them have. They sell them at Cabelas. We did not go that route because we got the Rides for half off but I do wish we had a better kayak for the dogs.