No, you could do this quite quickly.
Because a “breed” is pretty much whatever we humans decide to recognize as one. There are no real biological or species type differences there – your new breed of dog or cat would be fertile with other existing breeds.
So if you have the money or influence to get a lot of humans to recognize the new “breed” that you’ve produced, then you have created a new breed.
For example, in horses, the National Show Horse breed is basically a 1/2 Arabian, 1/2 Saddlebred cross. No doubt such cross-bred horses have existed for a long time. But some years ago, a group of breeders (with money & power) decided to give them a new name, and call them a new breed, and they put money & effort into creating a registry, classes & shows, etc. for them. And so they made this into a new “breed” of horse.
That’s really all it takes.
Cross existing breeds of dog until you get what you want, give it a name, and use your money & influence to get the American Kennel Club to recognize it, and boom, you have created a new breed.