I haven't been in the Lake District for many years, but as I remember the weather is just what you said: "Possibility of heavy rainfall".
I stayed in Kendal - it was a school trip, so not much choise. If I were to go there again I'd aim for the middle - Keswick? Anyone local who can help?
Lake District looks rather like a scaled-down Norway in fact - are you sure you wouldn't like to visit "the real thing"?