Honestly there is no way to wash a building without getting cleaning chemicals on the brick.The trick is to pre-wet the brick before you apply your chemicals,no matter what chems you use you'll still have to rinse the brick.(rinsing the brick doesn't require high psi)
We don't just wash trim,we wash the whole building(or at least rinse it before,during and after)the price is the same.Often times when people want the just the trim cleaned they think they are going to get a lower price.
Never heard of "mildew check" chlorine diluted to your needs would do a great job.