Do what you like but if they cross over winter it will be dark the whole time.
You are supposed to be crossing in 'the off-season'. I made it an event. "Yesterday, the sun barely rose above the horizon before setting again. Today, it doesn't rise at all." And later, the sun finally dawns once more, hopefully symbolising something or other.Compared to temperatures of sixty below, mere perpetual darkness shouldn't be much of a problem for magic-equipped adventurers.
As others pointed out Book 3 should take place over late fall -winter - early spring. I started Brinewall in mid summer as the adventure and the trek from Varisia to the Crown of the World would in of itself take several months. Not to mention the time needed to prepare and outfit the caravan for such an expedition. So by the time they're ready to start the crossing its already at the end of trade season. So yeah winter darkness for much of the journey depending on whether the group decides to stop and stay for a protracted period of time somehwere.
From practical play experience, I recommend setting it in Sarenith or Erastus, since with all the traveling done in the AP, you'll otherwise miss the real polar night when traveling over the north pole. That sadly happened in both my JR campaigns, where I started the campaign in Arodus.
I went the other way and crossed the Crown of the World in summer.
Okay, I *may* have forgot that it was supposed to happen during the winter. It's only later I realized it was easier to navigate by northern stars than by an unsetting sun.
Still, it made for different events. More snow blindness, less Howl of the North insanities. No good luck auroras, but increased likelihood of meeting herd animals and find some arctic vegetation.
As it is, the group entered Hongal around September and are about to hit Kasai around January. Winter is quite settled in, and the general populace will feel even more miserable.
travel time in the books doesn't match the core book, for overland wagon speed, so I basically ignore the listed travel times and go with my own.
my start date was 14 Pharast 4712
arrived brinewall environs 20 Gozran
arrived Kalsgard 15 Sarenith
left Kalsgard 29th Erastus
Arrive Illquiat 3rd Kuthona
Arrive storm tower 25th Abadaius 4713
Arrive Necropolis 31st Pharast
Arrive Ordu-Aganhei 12th Gozran
Arrive House of Withered Blossoms 19th Sarenith
presently they're at the 27th of Sarenith having defeated Munasukaru, and are planning their between book - downtime...retraining, teleport back to Kalsgard to sell loot, they're planning about 2 months (my guess, 3 crafters) of crafting while guests of the Kami in the forest of spirits