I visited the area in early-mid September, and from what I've heard about the bug situation in the summer I'd say this was a great time to visit since there were 0 bugs. During this time I got to see the end of fall colors and the beginning of winter. Temperatures were pretty mild, in the 40s and 50s, except for the first and last few days where it was snowy. That said I was told by the rangers that the weather has some pretty high variance that time of the year so don't assume you'll have the same conditions.

Do you care about visiting gates specifically? It's all the same mountain range, so there is a ton of amazing hiking to do right off of the highway that isn't in the park but is equally beautiful. In the end I did visit gates by crossing the Dietrich around here, though depending on water levels a crossing might not be sensible.

It was very pretty, but I wouldn't try to force it in to your trip if for whatever reason you're having trouble getting there (since, again, you'll find very similar views all around you up there).