We've been in the same position as you and I spent about about 18 months looking into it and eventually we decided that, although I do strongly believe in adopting animals wherever possible, we would go down the breeder/puppy route to give ourselves the best chance of assimilating a dog into our child /cat household.
On looking into rescues it felt as though just having young children wouldn't have been an issue with all of the rescues, but the combination of cats + children made it almost impossible. I looked for almost a year at several different rescues and didn't see anything that they would recommend for children + cats. I spoke to a couple of dog behaviourists who recommended a puppy for the best chance of success with the cats. Our local Dogs Trust centre did tell me that the only dogs they would consider homing to us would be puppies and that they do get puppies in quite regularly at their centre, but that they go so quickly that you would need to be able to call into the centre in person at least every week or every few days in the hope that a litter may have come in - puppies never make it on to the rescue centres websites and the Dogs Trust don't hold waiting lists so you have to be willing or able to physically go there regularly over a period of time (this was pre-Covid, I don't know how they are operating at the moment). So this may be an option for you.
We also looked into rehoming a Trailhound, and only really decided against it because we felt a smaller dog would be better for us. Lakeland Trailhound Rescue rehomes trailhounds, either those who've reached the end of their competing life or younger dogs who just haven't made the grade. The dogs may have never lived in a house, so will need to get used to this. However, many/most have been brought up in environments with children and often lots of other animals around - they've often lived on farms or multi-pet households already, which is reassuring if you have cats. The lady I spoke to at the Trailhound Welfare was really lovely and I've read good things about the organisation - it's definitely something I would consider if you're thinking of a larger hound-type breed. www.trailhoundwelfare.org.uk. They seem like lovely dogs.
I've often read that some of the smaller rescues or the breed-specific rescues can be more open-minded at considering people with cats and/or children. Some of the breed-specific rescues in particular may know more about the history of the dogs and which have lived with cats in the past.
One thing I did feel wary about was some rescues I came across (particularly those rescuing from overseas) whose "cat tests" seemed very very brief indeed. I wouldn't rely on a "cat test", or at least I'd ask lots of questions about what the test actually entailed. We did actually attempt to adopt a dog who had been living in a foster home in a multi-dog household with one cat, so he had been deemed by the rescue organisation to be "cat friendly" and we were reassured by this even though with hindsight there were behavioural warning signs that had we been more knowledgeable we would have recognised as red flags (I got told this by the dog behaviourists I subsequently spoke to). It almost ended very badly for one of our cats, despite the attempt at keeping them apart initially and following all advice re. careful/distanced intros. One dog behaviourist told me the dogs' own instincts were suppressed in the foster home (which it had only been in for 2 weeks) because as a newcomer it would be submissive to the dogs already in the house and would follow their lead regarding behaviour towards the resident cat. I'm just mentioning this to say, be very careful to find out what you can of a rescue dogs history re. cats and don't necessarily feel reassured if a dog has lived with cats in another environment e.g. a foster home, or has met a cat briefly once.
Good luck with your search.