Our pets are NEVER outside when we are mowing the grass or doing any trimming of bushes or trees. It's just too dangerous for them and for us. That's our rule and I'm sticking to it. LOL
It isn't just that the mower throws grass, twigs, and other things out so quickly that our dogs might get hit by flying debris but how can we possibly do a good job moving if we need to keep an eye out on our pets in case they get too close to the mower? The machine can hurt your pet and the machine can hurt you if you have to dodge a moving target in the yard.
I would also be very concerned about the neighbors mowing with their pet running around. Unfortunately, I don't think there's anything you or I can do about it.
Someone else might have other ideas. Good luck and it's obvious that you know what you are doing with your pets, good for you
!!