Or, OP doesn't care enough about integrating her children with their wider family to address her fear of dogs
OP and her dh seem to care enough to invite all the relatives over to their house every weekend. But they won't come without their dogs. Doesn't sound like she doesn't care about integrating with the family. I can totally empathise with this being a deal breaker. A 6 month baby and a toddler in a house with 4 unfamiliar dogs sounds like a recipe for trouble. Can't put the baby on the floor for a start.
I've contributed to this thread a few times and I have lots of dog owning friends and relatives, but in all honesty only one of them in 40 years has actually asked to bring their dog into my house. It was the retired couple next door who never go anywhere without their dog and I invited them to a few neighbourly drinks and nibbles on NYE. (alongside a few other neighbours, a dozen or so in all)
They turned up on the doorstep with their medium sized mutt and I looked at them quizzically. They came forward to step in the door and without thinking I reacted by half shutting the door. It wasn't a good optic, I admit.
I just said through the gap "Why have you brought the dog"?
They said they couldn't leave it alone, it would cry and bark. I said 'there are a dozen people in here, the dog won't like it"
They said they would come in and if the dog was barky and didn't settle they they would leave. I had to say I didn't really want to disturb things as they were, and didn't know how the other guests might feel (half of them have dogs of their own that they didn't bring!). They then said the other neighbours knew their dog because they see them out walking. So I had to resort to pure honesty and say I didn't want a dog in my house.
At the time they mumbled a bit and one of them came in and stayed for a couple of hours, then swapped places with the other who went home to be with the dog. We'd been neighbours for 20+ years but they became decidedly cool for a year or two. They were so mortally offended by me that it was difficult for them to speak to me after I 'dissed' their dog. I mean, do dog owners have to take it so personally? I like these neighbours, they are lovely and friendly and I wouldn't ever want to upset them. But having their dog in my house, let alone having a dog significantly change how the gathering might go - you can't sit here, can't sit there, don't go near the dog it's asleep, don't feed the dog, don't stand on the dog, there's a dog around so don't step backwards without looking. . .that's way beyond what I'd do for my nearest and dearest.
Having said all that, and answering the yet unasked but anticipated question.
"Would you have shut the door on a baby or toddler?"
No, I wouldn't. Not ever. Babies and toddlers are human beings and are welcome in my house day or night, whatever the reason and no matter how taciturn they choose to be. And particularly on New Year's Eve, I would assume that the parents really wanted to come but had no childcare support whatever so chanced their arms and brought the babies. Some might think children at a New Years Eve Party is inappropriate. I would think it might be their only option of getting out and socialising and I would have gone out of my way to make them extra welcome and glad that they could engage with us and felt comfortable doing so. I would go miles out of my way to make parents of babies and young children welcome in my house. I would make all rooms and beds available for them to sleep, if that became an option. I would go out of my way to prepare food that they could eat.
But not dogs, never dogs, because dogs are, well, they are just animals.
Animals can be trained to be left alone for a couple of hours.
If you can't be arsed to train your dog then don't expect everyone else to accommodate your lack of commitment to being a responsible dog owner.
People who compare dogs to human babies and human children are, in my opinion, completely lost to reason. You just can't argue with stupid.