I really, really don't know why Adventure Cycling has the Great Divide Trail going north to south.
Not only does it not make sense weather-wise in Montana, it doesn't in New Mexico, either. Even when spring comes a little earlier to Montana, May and June are the wettest months of the year - plus high elevations still have to melt out. Soggy is the operative word. Then in July August, the monsoon cycle hits New Mexico with heavy rains making some of the clay roads a nightmare.
Add to that the prevailing wind in much of the Rocky Mountain West is southerly - SW in New Mexico and Colorado, S & SW in Wyoming, S & SE in Montana. Granted that wind is not as much an issue as on pavement - and that you will have forest cover for much of the ride. But there are also long exposed stretches - like the Red Desert in southern Wyoming and lots of places in New Mexico.
Then there's the relatively minor point of biking into vs biking away from the sun.
Although I have never biked the GDT, I live in Wyoming and have toured extensively on dirt. I have ridden many sections of the trail. For example - the first time I rode the stretch from Eureka, MT to Polebridge was back in 1990. The road was closed because of snowdrifts, but I was able to get my bike over them and and empty riding. But I camped primarily in lower elevations - even that was soggy.
Most people have a mid-June to mid-August time frame.
East of Jackson, WY you have bentonite soils that turn into cement when wet. I have had to carry my bike downstream because I couldn't even walk the roads - let alone ride them. Yes, the weather is supposed - supposed to change next week, but the higher elevations are getting snow today and tonight - up to a foot. That's on top of record winter snowfall. I don't expect the high country east of Jackson to melt out until mid-July at the earliest.
Have you checked Snotel?
There's still three feet of snow at Stahl Peak at 6000 ft.
Three feet in the Pioneer mountains.
Four inches of new snow at Many Glacier - could be a foot.
I'm sorry to be the bearer of bad news,
But I have begged Adventure Cycling to suggest a reverse direction for the GDT -
Only to get a dismissive reply - i.e. "People have different opinions."
I could show you tons of weather data as well of twenty years of personal experience
to suggest otherwise.
I'll leave you with two precipitation graphs -
Notice that Helena, MT has it's peak June 1 and a trough August 1
Grants, NM has a low point on June 1 and a peak on August 1.
Feel free to contact me.
I'll share any info I have.