Look up the historical data for say Henningsvaer on yr.no - I'm sure they have excellent info there. I've been in the Norwegian arctic in mid to late August a few times and have had wonderful weather BUT a month later it will have cooled down more and the days are a bit shorter. Further south and inland in Lapland that is considered one of the most beautiful times of the year - much of the birch will be golden and colours in the Nordic autumn are wonderful - it's one of the more profound experiences of my life at least. But of course Lofoten is by the sea, so just like Scotland while a week of golden, midge free perfection is possible with blue skies and crisp temps, I'm sure a week of low cloud, SWerlies and constant rain is also!
Much of the climbing on Lofoten is not high above sea level so I don't think you'd need to worry about first snows, although I'm sure that's common inland on the mountains.