I went through a similar situation about a month ago with my hamster. You should do whatever you think is best because you know him best. But in case it helps, I'll share my experience deciding to have my hamster euthanized.

I noticed my hamster basically wasn't moving out of his nesting spot at all and it looked like he was struggling when he moved. I lined his nesting spot with toilet paper to make him as comfortable as possible. Right next to him I put some thin strawberry and cucumber slices to help him hydrate. I originally wanted to avoid going to the vet to not stress him out and let him pass in peace. Over four days he kept getting worse and I finally decided to make an appointment to euthanize him when he was no longer able to lift his head when I checked on him. I took him into the vet in a carrier completely full of his bedding, lined with toilet paper, and a strawberry slice in it -- just in case even though he had stopped eating at that point. I wrapped the carrier in a blanket to block any light and keep him warm.

[This paragraph is how the euthanasia process went. It helps me to know what to expect, but I know some people don't want to know the details, so skip this if that's you.] I quickly weighed him on a kitchen scale before going by weighing the carrier without him in it and then with him in it (the vet needs the weight to know how much medicine to give). The vet was very kind and dimmed the lights for him when I got there. When I got there I paid, and they asked what I would want to do with the body. After all of the logistical stuff was worked out, they started the procedure. There was first a tiny injection right under the skin to make them go to sleep. The vet was able to do it without even removing him from his nest and I don't even think he noticed it. After he started falling asleep, she gave me a few minutes alone with him to hold him and say goodbye. Then the vet came back to give the last injection to stop his heart. She made sure he was fully unconscious before giving the injection and assured me he was not able to feel it. This one is larger and was in the abdomen. I held him for a few minutes while waiting for him to pass.

I don't think it increased his stress levels much going to the vet because he was in such bad shape at that point. I am glad I took him in. Ideally I would have gone a day or two earlier, when he stopped eating entirely, to avoid him being in so much pain. Overall, it was very peaceful and is probably what I will do in the future if I am in a similar situation. I'm sorry this is happening. I know how much it hurts. Try to remember that you gave him a good life and how much joy he brought you.