I had 2 children, a newborn and a 19 month old. My husband had left us, we had very little money, no support and no transport. I was struggling having just had a caesarian section... and I had 2 dogs.
Cassie was already older and she died when my children were quite small. Monty was only a year or so old. He joined Cassie three and a half years ago, aged 12.
I was tired, exhausted, in pain, desperately pushed for time, no car to nip to the shops, no husband to feed the babe whilst I cooked for the toddler here. But I couldn't have given my dogs up for the world. I owed them, they relied on me, and they weren't just disposable items to be discarded when it didn't suit me. It wasn't their fault. Besides, taking the chance to walk them, with a babe in a sling and toddler in a buggy, or both in a double buggy, was my chance to unwind, to stop being a cook, cleaner and nurse. It kept me sane.
So, my thoughts are these, for what they're worth. You are being unselfish by sending him to Lab Rescue. I admire you for that. BUT... how much will you miss him and how much will you regret not finding that extra half hour or so in a day when he's gone?
As time goes by, you establish a routine, you tell DH you need some "me time" to walk your dog alone, you'll find that the time you spend with him naturally starts to increase again. Children grow, their demands change, workloads change. They become less hard work.
No-one is going to castigate you for having less time for your Lab now that you have two youngsters. We know that it won't be forever. But if you let your dog go now, that will be forever.
Purely my own take of course AMWOW (god that should be my name!), and I don't know the ins and outs of your life, but all I can suggest is that I managed it (have had at least 2 dogs of my own, have fostered dogs and gone through cancer all as a lone mum). so I'm sure you can too. Perhaps you can get that husband of yours to care for the babes regularly while you take a breather out walking with your dog. Whilst indoors, compensate with more mental stimulation for your dog, in the form of chews and Kongs, stroke him and talk to him each time you pass him whilst doing the chores, play fetch/hide and seek a toy/treat whilst you're sitting having a coffee.
Hope that I've offered food for thought and reassurance that it's not impossible to have both cared for children and a happy dog.