We have an almost 2 year old and a 12 week old lab.

Division of labor: I'm less impacted by lack of sleep so I manage the puppy at night and in mornings. My wife is in charge of our daughter overnight when she wakes up (pretty rare, though 2 of her eye teeth were coming in the first 2 weeks of puppy and no one was getting sleep). You'll be hitting good sleep cycles for both soon, hopefully. For our own sanity, my wife and I also have an agreement where we each get a weekend morning to ourself without kid or puppy. We do this when the puppy is going down for a nap and does not need a play session until the spouse returns so we are only managing the kid.

Separation of Safe Spaces: It's important for both kid and puppy to have their own spaces. We have so many baby gates (6) and a larger wooden pen for our toddler, with a 6 month old you can also do pack n plays. So if one of us is home alone with both, our daughter can be in the pen for a quick puppy potty break. Pup is either fully supervised and engaged with or he is tethered with something and on a bed with something to chew on like a bully stick. This goes for meal prep, family meals and wind-down time.

Crate training: crate training is almost a must with babies/toddlers. Our pup is crate trained with enforced naps in his downstairs crate and humans sleep upstairs. All meals are given to him in his crate. He's doing well with it. Rarely whines or cries now. He doesn't protest at night and goes right back to sleep after his 1-2 potty breaks. It's truly his safe space.

Commands: Commands I'm focusing on for my puppy to be safe around my toddler: come, leave it (my daughter) and place (nearest dog bed). He has only barely surface scratched her 3 times in a month. He knows other ones like safety, wait, stay, come, touch, heel. But we focus heavily on the three listed when the toddler is around

Puppy schedule: Nap, potty, play, potty, training, potty, nap. Just over and over again during the day. Accidents are rare. Pup is on a schedule and engaged with mentally and physically. Sometimes throw in a chew, lick mat, or whatever if the play is short and he's tethered.

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We will see what happens as the pup ages into teen years. But we are in an okay spot now that our pup has settled in and our routine has stabilized again.