Bain:
Leaving Vancouver May 20 is no problem, winter will be well and truly gone by then, and the temperatures will be 'mild' before you get into the mountains.
Prior to June 1, you run a bit of a risk of encountering some snow as you go across the mountains. But, the worst possible case would be that you get stuck at a hotel or lodge for a day or two. More likely, you'll get some overnight snow that will melt by noon hour - you will then have wet roads in the afternoon.
You don't mention how long you plan to take to do the trip. If it is going to be a two-week trip (putting you in the mountains in late May), the above applies. If it will be a two-month trip (putting you in the mountains in early June), then you don't really have anything to worry about.
Having said all that - If I was doing the trip myself, I would try for a June 1 start, simply because the weather would be warmer and thus the riding would be more pleasant. Mountains can be 'shady' places, and as a result, the air temperatures can be a bit cool in late May.
Michael
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