For the most current status of that bridge I'd give the Boise NF a call. That specific location is on the Idaho City RD (208) 392-6681. It's been a few years, but I think you can easily bypass that bridge by taking routes a little (5-10 miles) to the east towards Atlanta. While you're at it, Atlanta may be worth a stop to check out the hot springs and/or local bar? Atlanta is an old gold mining town with just a few full time residents left, but it does have some rustic character.
September in Idaho can be rather smokey with numerous wildfires so just plan on the possibility of having to re-route parts of your trip.
2001 Sportsmobile RB50 7.3
West Central Mountains, Idaho