You can take a Greyhound bus from Denver to Edmonton. It takes about 2.5 days and might not be what you want but they are probably one of the few options that can handle a tandem.

The snow so far has mostly been in SW Colorado and yes, it has been heavy. You are probably right that some passes may still be closed at the end of May down there. The rest of the state hasn't had nearly as much snow and it's been a pretty light snow year so far. That can all change come March/April but I'm guessing that SW Colorado will continue to get the bulk of the storms this year.

Almost any pass with a paved road will be open by June with the possible exceptions of Independence pass and Trail Ridge Road. Those usually don't open until June in a normal year. A lot of the roads up high will be very wet and sandy in May and June and you can still easily get snow then as well.