Molly (my Berner) is just 2 and my daughter is 9 months. I would still consider Molly a full out puppy besides the mouthing- her energy is still high and she's super playful.
She is fabulous with my daughter. I've never worried about them being together. Molly really seems to get she needs to be gentle around her and she loves being with her. my daughter will pull her fur the odd time and her tail just wags.

That being said I've done A LOT of training with Molly in the past year and it's taken a lot of time and work on my part. We brought her crate up from the basement (as it was put away for about 6 months) and I began sending her to her bed and exercising 'wait' with her. I've also done A LOT of work around her food dish and have since she was a pup.

In short, I think a Berner is perfect in a family environment, as with any puppy you're going to get you'll need to be on top of training and it will be a lot of work. Your kids are at ages too that you'll have to get them on board with your training so they don't reward your pup for undesired behaviours.
I know Caesar's controversial but he has chapters in his books about training dogs around kids and I did find it helpful.

I think big dogs get a bad rap around kids. I heard so many times 'are you getting rid of your dog now that you have a baby' when in reality Molly is a very gently dog who has never barred her teeth or shown any sort of agression towards a person or dog.

Bernese dogs are gentle, cheerful, and they love children. this kind of breed have a happy & cheerful attitude which makes a great family companion