How do you get the dog to hold the tag close enough to a reader for it to be read? The reading distance is pretty small. I'm also not sure how you work out the ten seconds either. You'd have to figure out if the reader kept sending a tag id all the time it was in range.
I don't think a servo will be up to the task. A linear actuator might be better. Do you leave your patio doors unlocked at night, or will you need to unlock them too?