Installing a pet door - no ideal location - support beam question
The image you provided from the internet is the typical solution. You state concern of the king stud for the window framing being too close to the exposed stud in your included photo to add an additional jack stud for the header for your dog door. The addition of a jack only narrows the opening one and a half inches. You need to determine the required rough opening for the pet door. If the rough opening is not at least 3 inches less than the current width between the two outside studs , you will need to get a smaller dog door , a smaller dog, or both! Good luck.