Solution 1:

I signed up for paid support and here's the answer, in case it helps anyone else. I had to edit the trust policy on the IAM role like so:

{
    "Version": "2012-10-17",
    "Statement":
    [{
        "Effect": "Allow",
        "Principal": {"Service":  "rds.amazonaws.com"},
        "Action": "sts:AssumeRole"
    }]
}

Then I was able to add the option to the option group and the restore worked. Note that the trust policy is called "trust relationship" on the web console.

Solution 2:

Below is the policy for a role with the permissions that should work:

 {
  "Version": "2012-10-17",
  "Statement": [
    {
      "Sid": "Stmt1472672338000",
      "Effect": "Allow",
      "Action": [
        "rds:*"
      ],
      "Resource": [
        "arn:aws:rds:us-west-2:123456789012:snapshot:ms-sql-rds-final-snapshot"
      ]
    }
  ]
}