During my brief stint as a cycle tour guide, I recall cycling into Salzburg with a scattering of snowflakes in the air in late May. I also recall leaving Krimml in very early June of that year with fresh snow starting to stick to the treetops just above the hotel, and very cold rain mixed with a little snow at the hotel. Yes, it could be cold, and it could be wet. The Koenigsee and the entire Berchtesgaden and Salzburg area are very pretty, but I wouldn't count on good cycling weather anywhere near the Alps in April, May, or early June.