Ohhhh I feel for you I really do!
We have the most amazing Labrador (5yrs) and now A beautiful Ds (6 months). Seeing them together is amazing and I know that's what you're envisioning
But the reality of those first 6 months with our Labrador was a whirlwind. Sharp needle like teeth and claws with no control. Having to teach him how to be gentle and bite control.
Coming downstairs to poo and wee in the mornings while he learnt to toilet train. Getting up in the night to let him out (we slept downstairs with him for a week but are soft dog parents!!)
Then at around 6 months he hit the teenage phase and would have mad bouts of energy where he'd throw himself around and we'd have to shut him in a safe space to learn to calm down.
He is the most amazing dog now but I'd say until he was 18 months he was still learning and wouldn't be totally reliable on walks and I'd have to keep an eye on him.
Now I cast my mind back to those first few weeks with my DS and it was a crazy, horrible, amazing, life changing, depressing, exhilarating whirlwind, I would not have wanted a puppy as well!
DS at 6 months is now reaching for DDog and actively wanting to touch and stroke him but as a baby, he doesn't understand being gentle. We have to be super careful and watch all interaction and luckily DDog is a calm and gentle Labrador but I would still never take my eye off them. I don't know how I'd cope if he was still in the puppy phase!!
I know you've probably taken the advice now but perhaps look into Borrow My Dog or sites like that where you can take a dog for a walk and sort of foster it alongside the owner?
Best if luck with the baby!