Unless one is very lucky, I don't think there is one start date from either N or S that will allow one to avoid potentially nasty conditions on the GDMBR - too much accumulated snow (Colorado) or impassable mud (New Mexico). Leave too early, you'll get the former, leave too late (southbound) you'll get the latter. Fortunately, the ACA maps provide paved alternatives to the mud: if fact, Tour Divide even uses the paved alternative from Cuba to Grants.

If you are willing to pack for inclement weather (potentially snow at higher elevations; otherwise, torrential rain), and are willing to take the alternatives to avoid impassable conditions, you can leave Banff or Roosville from mid June to the end of July +. This year, I left Banff on June 24 and arrived in Antelope Wells on August 25. I don't recall any accumulated snow along the route, but I did take pavement from Cuba, through Grants and even through the El Malpais National Monument - all to avoid the mud. I think you can get a good idea of the nature of the window of feasability, both N to S and S to N from the journals on Crazy Guy.

BBT: If I had a month starting July 1, I'd do the Canadian section - especially the new Flathead Wilderness routing - but then veer off at Polebridge into Glacier, go to Waterton, "figure 8" back to Banff on the GDMBR (yes, it's worth it to do that section twice) and then go up the Ice Fields Parkway. If you wanted to stay on the route, I'd just continue from Polebridge and quit when I ran out of time.

This year I went with a group that soon parsed into compatible "teams". Next time I will go alone with my SPOT, but I'd expect to hook up with other riders if / when it were mutually acceptable - especially through the lonelier bits like the Great Basin.

If you have a hardtail, I'd go with that rather than the LHT. Panniers work fine because there is very little single track. Besides, in the north you'll find it easier going over logs if you don't have a trailer. Usual caveates, yada, yada...

S to N: From a climate standpoint, it's doable starting from as early as the first week of June, but again be prepared to bypass blocked passes in Colorado. The advantage of an early start from the South is that one avoids the monsoon season of New Mexico - so less mud - but water is still relatively plentiful. One then encouters warmer weather in Montana and Canada. The downside is that some nasty decents N to S become even nastier ascents S to N.