Somehow I received the quest from the priest (Dawnmaster Kreel) in the Lathander temple, even though I’m not a cleric. I did everything as expected: talking to people in the cult’s area, talking to the diseased ones, to the Avatar, killing Unseeing Eye, returning the rod, killing remaining members of the cult.

But when I went back the the temple and tried to talk to Kreel the dialogue says he had nothing to say to me. What’s even weirder is that he was outside the temple, but when I went inside I also saw him there. Could this be a bug? Is there anything I can do?

I vaguely remember a bug with there being two instances of Kreel, but he's not the one you need to talk to. You need to talk to High Watcher Oisig in the Temple of Helm.

which temple asks for your help depends on your alignment... your quest list will say which one to talk to