No commercial vehicles are allowed on the Icefield Parkway. My guess is the system is equating your outfit to a commercial one and trying to divert you at Calgary. There are no problems with going that route.
You should be OK May 1, just don't push it. By that I mean if there is a storm or threat of a storm sit it out for 2-3 days. Rain at Calgary or Banff would probably be snow further up. You want to do the drive on a nice day not only for the safety aspect, but it will be one of the most scenic segments of your journey.
There may well be snow encroaching on the shoulders particularly at higher elevations. If so be cautious of black ice from the thaw freeze, particularly in the mornings.
There will be a limited number of campgrounds open and no water to sites. This will be the same both in and out of the mountains.
The mountains IMO are much prettier in May than in summer due to the amount of snow cover. There is also a greater chance of seeing wildlife.
Going June or Aug would probably give you better weather but it is still just a chance. I have been at Lake Louise July 4 when it snowed all day.