I mean, if this should happen to me, I want to undergo euthanasia as soon as possible. If I am already dead, I don't want to unnecessarily suffer. So my question is, did he not want the euthanasia or was it not "accepted" or why he had to suffer so much?

The first person the demon core killed, Harry Daghlian, notably allowed the doctors to study and record information about his deterioration due to radiation. I believe Slotin had a similar motivation - that at least, even this slow, painful death could provide valuable information to doctors and scientists.

This was the United States in the mid 1940s, I doubt euthanasia (or even assisted suicide) was much of a thing back then. Plus, as someone else mentioned, there was also the scientifical aspect of being able to study the effects of irradiation.