Well, beyond the fact that proposing marriage to a dog
(by which one promises to spend the rest of one's life with that dog) probably means he's accepted his dog nature...
There's these two
strips which show King realizing (and finding some surprise) that even given the option to return to being human he doesn't want to. Then of course there's the ever famous "I will move heaven and earth to get you (Bailey)
" line, which again seems to indicate that he is willing to do whatever it takes to remain with Bailey (i.e. actively working to remain a dog).
There isn't really a specific point where King says "I've accepted my dog nature" or anything like that, but it becomes apparent slowly but surely throughout the story that he's accepting the reality of the situation and then accepting that he is okay with that reality.