It sounds like you have some reasonably compelling mitigation and I'm minded to say you should put it to the authority.
However we know Birmingham does not extend the discount for CAZ cases, meaning that if you make representations and they reject, they will demand the full penalty.
That being said if they do that, there is then no further cost to appeal to the Traffic Penalty Tribunal, and to date Birmingham Council has not contested any of the appeals where I've acted as a representative.
If you want to challenge the PCN, I think we could put something reasonably compelling together.