Lucy, I have been in a similar spot just a few weeks ago, I took Darla to the vet though, I had the same concerns you had about having the last moments in my home. It doesn't matter if your house is small or large, a sensitive soul such as yours will always...well, you know it will enter your mind anytime you see that part of the house. Darla was not too freaked out about going to the vet, but either way you take Riley in if needed for other visits, even if he is scared, this time will be no different, and the first injection will put him at ease quickly; wait in the car until they are ready for you, distract him with yummy treats, he will be ok. I too experienced this for the first time, and my vet and vet tech were really good about explaining what to expect, they told me some dogs will sound like they are gasping for breath, but it's just a reflex, they are in no distress, like someone else mentioned, some let go of their bowels or pee, but not all do.
Darla was shaking and panting at home, and at the vets, because she was in pain, so when that shot took effect, she relaxed, and I could feel her relief. She had refused to lie down on the blanket, so I held her and talked to her, the vet tech held her back end, and we slowly eased her to the floor once she relaxed. Everyone reacts different, it's ok to cry, have a friend with you if want, I wanted to be by myself, and I went to work afterwards, to have a distraction, and because I could not go back home right away, do what feels right for you. Riley is a lucky boy to have you for his mommy, I know it sucks having to be the grown-up....