First - the date - I backed up from October, not seeing the May 1. Still May 1 is early.
There's an excellent website of Western U.S. climate - NOAA Climate Summaries | Western Regional Climate Center
If you look at records - there was a cold snap with highs in the low 40s and a few inches of snow in late May in 2011.
There is almost always a cold snap in May in the interior West - in Wyoming it can be Memorial Day or later.
I was paid by the Census Bureau to ski into summer homes in the Bighorns in June 2010 to confirm any residents.
Warmer/drier years tend only to have a quick snap, but colder/wetter years can have pretty brutal Mays.
And this is supposed to be an El Nino year - esp. in the Southwest.

About south to north on the GDMBR - I actually think it is the better direction for a June/July full GD trip.
In Montana, there can easily be late snows and snowpack has not had a chance to melt out.
In contrast in New Mex, June is at the end of the dry season, by August the rainy season turns caliche to cement.
But the challenge for south to north in in the Colorado mountains if you hit them too early.
A week or two often means the difference between a doable, but brutal ride and a really wonderful one.

I've ridden the Rockies from New Mexico into Canada more than a half dozen times.
The GDMBR has some wonderful and some not-so-wonderful sections.
Also, 250 miles per week is really on the low side for someone your age and strength.
I was doing 350-400 and I'm an old geezer. Plus I was hiking into wildernesses.

Have you considered zigging southeasterly towards Taos before heading north?
There are some incredible ancient Puebloan sites - esp. Chaco Canyon - remote.
Hit Chama or Taos and then turn north??

Anyhoo, that's my 5c.
Best - J