I am not sure if it is an option where you live but here there are vets that come to your house and put your animal to "sleep" there. Some people prefer it as their animal is in their own calming environment, on their own bed and not distressed. They also let you spend as much time as you want with your pet before doing it.
The saddest article I ever read before was from a vet who said the worst part of their job was when an owner would take their pet to be put to sleep, sometimes the owner wouldn't want to be in the same room and the animal would start panicking and looking for them

. So never go out of the room, be there for your sweet pets.