I've done the trap dike many times in Winter with a fairly heavy pack (rope, skis, etc, on pack), although not with a big overnight pack.
In summer conditions, having a heavy pack would make the "second waterfall" section of the climb a little "balancey." It might be a bit uncomfortable; you might want to bring a short (50' or so) length of rope so you could leave the pack at the bottom of that section and then haul it up once you're back on easy ground.
Oddly, in winter conditions with a good ice axe planted, the dike is a lot more secure and easier with a pack.
As to early November, unfortunately that's a tough time for climbs like that, because it's an "in-between season" that often features the worst possible conditions, such as verglas. I haven't climbed the dike in verglas conditions, but it would be a little tricky. I would bring full ice gear as for a technical climb, and expect it to be harder than in mid-winter. You might be pleasantly surprised; it all depends on the weather. But I'd be prepared. If it had been unseasonably warm and dry for a few days, I'd say try it. But if it's typical November, I'd treat it as a technical climb. And if you find verglas in the dike, stay off the slide. Verglas in the dike would be manageable if you're a decent ice climber, but on the slide it would be impossible.
TCD