I did it - picking up a 8 wk old lab x cocker at 7.5 months pregnant. We knew my maternity leave and then my partner being a SAHD at the end of maternity leave for a couple of years would give us a training window we'd not have pre pregnancy. Any RL discussions and MN threads I saw advised against it and we'd not have got one this exact timing but a litter we liked the look of came up - accidental mating of 2 x working dogs (we are training as gundog) with good hip, eye scores etc. It is hard work and many early mornings nappy changes coincided with puppy weeing, grabbing dirty baby wipe, bring in clumps of grass from the garden etc etc.
My personal struggle was having to to walk puppy and our older dog separately as pup was terrible on the lead and doing that in the rain with a grouchy baby in a sling was no fun. Pups need short walks (5 mins per month of age) but for ours that is non negotiable, he is a high energy mix and a nightmare if not exercused physically and mentally.
You need to train the pup from the start and with absolute consistency - for eg when breastfeeding pup is told to go to bed or he brings us toys, comes to sniff baby feet etc and can cause her to unlatch. We crate trained our older dog and m while not popular with everybody, I find crate training an absolute godsend for when baby has to have my attention.
When making a decision bear in mind that with a pup when baby sleeps you wont be able to nap as other mums of one may be able to - your pup needs affection and training.
I have also been lucky with a smiley baby that naps well and self sooths, friends have wailing babies that often only sleep on them. I could well have been in pieces if my baby was like that. She (baby) is only 3 moths old and I am under no illusion that all will be smooth.
Remember a new pup will tie you to the house (trips out need to be short) or to dog friendly outings.