I think just by luck more dogs aren't killed in TV. Everytime I get my mail, there is somebody with their dog sitting on the front seat of their cart as they pull in or pull out. They sell these on Amazon, if you got seat belts for yourself in your cart, this will work.
Here's a whole page of harnesses for sale
https://www.amazon.com/s?k=dog+harne...ref=nb_sb_noss