@puppyplans - I would suggest not completely ruling out rescues as the staff can often tell you if a particular dog gets on with cats or not.
Re: Poodles - an aged relative had one when I was a child and it was super yappy, (although devoted to her). Another relative has one now and it just wees everywhere all the time when people come round. (I had no idea that this was ‘submissive weeing’ but it makes total sense, so thanks for that info @Killingoffmyflowersonebyone ).
I know poodles are very clever and can make great pets, but there’s no guarantee with any dog of being trouble free.
The number one priority, as I think others have said, is that any dog gets on well with your cats. Also, do your cats like dogs? (I seem to remember a recent thread where cats had not taken well to the introduction of a dog.)
Having said that, another relative adopted two farm collies - could have been a disaster but they are adorable and the already resident cats just exist side by side - no friendliness, but no aggro either.
Good luck with your search.
PS: With resident cats, I would avoid any type of dog with a strong prey drive - even the nicest natured of these might begin to see kitty as a tasty snack or at least a fun toy to chase 🙀