Must have been going through and Could have been going through
Solution 1:
You are correct, there is nothing grammatically wrong with "could have", but the meaning is different. When you say "could have", you are talking about something that is possible, but not necessarily true. When you say "must have", you are talking about something that is true. Arthur Rodway must have been going through a painful illness. But while he was going through a painful illness, he could have also been going through other things, related or unrelated.