I recently got a kitten to keep my dog company. While the dog will sleep through the night, the kitten likes to wake me up around 3 a.m. by batting my hands or feet.
"While the dog will sleep through the night, the kitten likes to wake me up around 3 a.m. by batting my hands or feet."
Far as I can tell, you just wrote a perfect description of kitten behavior. I love them dearly but there are times when they can be a handful. They can be very, very busy when they're young. When Elliott was little, we used to try to tucker him out before bedtime. It sometimes made it harder for him to go to sleep, but it often meant he'd sleep for longer periods of time. All that changed with time. He's been on our schedule for years.
If all else fails, you could try making up a little bed for kitty in another room. Hope everything works out. He or she will likely be a fine friend for your dog.