Hello 1_travel_lover, and welcome to cruise-chat;
the weather in Alaska is really hit-and-miss; you can have great weather in May and rain or drizzle in July and August. But there is a reason the prices go up toward mid-summer and down again toward September: that is when you have the best chance of good weather (and of course demand is higher in mid-summer).
I'm looking at last season's brochure so the numbers will be a little different, but I see a $540 difference between a May 12 sailing and June 23 through August 4 sailings for a deluxe oceanview on Diamond or Sapphile Princess; but there are sailings on June 9 and August 18 for only $200 more than the May 12. I would not pay $540 more for a prime-time sailing; but I would pay $200 more for a just slightly less than prime-time sailing. Are you still with me?
In short, when you are paying that much for a cruise, I would spend a couple of hundred more for a much better chance of good weather.
Keep in touch; let us know what you decide.