So presumably 5b would be the problem here.
Hard to say. The bridge itself does no damage to the environment. It is hard to argue that increased traffic from people being able to use the bridge would create unacceptable damage given that people can access the area through other means. The bridge is a mechanism to increase personal safety.
What exactly is their concern: Is it the mountain goats?
You might argue that consent was not actually required due to:
"(2) Despite section 102 of the Act, a person may construct, rehabilitate or maintain a trail without the consent of the district manager if doing so is the only reasonable means of minimizing a risk to personal safety."
It is unclear. They may argue that this clause only applies in the case where you are in the woods and face an imminent risk to personal safety if you do not construct a reasonable way to get out of the woods.