Thanks cypher and TZFan!
It was a bit of patience and a lot of luck. I was At the exhibit for probably 15 minutes and she stayed in the den the entire time. No one else came by during that time. I had just started to walk back up the hill when Ember came out, stretched and then brought the kitten out.
I saw no sign of Ryan inside the main exhibit, so I would assume you are correct that he has been brought inside TZFan.
Ryan being separated makes total sense. Lynx fathers are not involved fathers. Having him on exhibit with her and the kittens would have been even more stressful for Ember. New kittens, her mate near by who could potentially turn on the kittens, and visitors gawking at her when normally she would be secluded and alone with the kittens in a private den not even seeing keepers for the most part. I bet she was a little stressed at first. Ryan being with her when she gave birth proves all the more to me that this was an accidental birth on exhibit. They would have separated Ember and Ryan if they knew the birth was imminent and had Ember off display where they could provide privacy while still monitoring and have an easier time intervening if necessary. Not only is Ember probably a little stressed but her keepers likely are too.
I would think unless they think one of the kittens is in trouble they will leave them until Ember either decides to bring them in on her own or they are big enough to follow her inside.