i'm surprised that some rescues have such blanket rules. We (Celia Hammond's) generally say it's fine to home to families with young children, although it will depend on the individual children. In fact I'm pasting below what we say on the subject:
We believe that it is beneficial for children to grow up with pets whom they learn to treat with kindness and respect. It is up to parents to ensure that their children do treat pets kindly, babies and toddlers must be supervised at all times when near cats, it is not fair on your cats if they are handled roughly and any cat that has it's tail pulled or is manhandled by a toddler or young child can bite or scratch.
If you are worried about being able to keep a watchful eye on your young children's interactions with a cat then it would be advisable to wait until they are a little older (5 years or more) so that you can discuss and explain to them how to treat a cat well.
We do frequently receive offers of homes from families whose elderly cat has recently died and are keen for another quickly, as the children in such a home are used to cats introducing friendly new cats should be straightforward even with young children.
There should be areas within a home where cats can get away from young children, stair gates are a useful aid in achieving this as well as a method of keeping the location of a cat litter tray separate from very young children, even a shelf for a cat to sit on out of reach allows the cat it's own safe space within your home. A cat flap also provides a cat with the freedom to come and go as he or she chooses.
Families living in a home without a garden where the cat will be totally kept indoors only should seriously consider waiting until children are older as unless the flat is particularly spacious there are usually limited places where a cat can find peace and quiet.
Being rescued many of our cats are not suitable for homes with young children, many are too shy, some have even had previous bad experiences with children in homes where they were mistreated. However we do also rescue normal, friendly cats and kittens, that would enjoy all the extra love and attention that children will give them.
I am sure there will be some rescues who would happily home to you and others which would maybe be happy for you to have adult cats, but not very young and vulnerable kittens. I would speak to some rescues before writing off the idea. And, as cozie said, if you particularly want a BSH then there are always specialist rescues.