What an exciting dilemma to have.
I would recommend a crossbreed, they are often much healthier and haven’t been beed for a single purpose so can fit in more easily.
Most dogs will want to harrass your other animals and so you will need to train them and keep them on a lead until you know you can trust them, with my first dog, a lab staffy cross this took a couple of weeks. My second dog, Jack Russel x Cocker Spaniel x Yorkshire Terriier, it was closer to 9 month.
In my experrience, rescue centres are notoriously reluctant to give up there dogs
I tried several last time with hundreds of dogs on their books and they all assured me none were suitable, I’m sure most could have been trained, like the puppy I ended up getting through a private sale, a mutt, from a smallholding. If you can make the rescue thing work then great, but personally I would keep my options open. If going through a private sale, look for someone with similar background to yourselves - if you want a family pet, a working pedigree may not be ideal.
It’s definitely worth the effort, I wouldn’t be without one.