Why haven't you paid to get the car out of storage? You are accumulating more storage fees and problems by not getting the car out of storage. If the court determines that the tow was valid, you'll end up paying more for leaving it in storage. Taking it out of storage will not adversely affect your lawsuit, but not getting it out may end up resulting in storage fees even if you win. In this situation, mitigating your damages is a worthwhile thing to do.
Note that your request for the court to order that you be given a temporary driving permit is something the court won't do. That request has no connection to the wrongful storage claim and the court does not have jurisdiction to grant relief not related to the claim at issue. There is nothing stopping you from doing whatever you are required to do to reinstate your license. It's up to the DMV to determine if there are any grounds for refusing your request for reinstatement of your license.
Note that the fact that you were not arrested and taken to jail doesn't seem relevant either. That, too, is unrelated to the towing and storage issue. If you think it's relevant, you need to explain why it's relevant to the towing issue because it's not apparent to me and it won't be apparent to the judge either.
Earlier you said that the police chief's wife owned the tow company, but your court petition says it was the wife of the officer who ordered the tow that owned the company. That makes a difference.
You don't address what the officer claimed was the reason for the tow. Why not? That is, after all, important to the issue of whether the tow was valid. The issue of whether the tow was valid does matter. You also don't make any mention if the other tow company was available to do the tow at the time. That matters, too. If the officer was justified in having the vehicle towed and his wife's company was the only available, he doesn't really have a choice in whom to call for the tow. That's significant because in that situation he's not making a choice to use his wife's company over another company that was available to do the job. Instead, he was calling the only company available to do the job.
Your petition leaves me with the impression that you've left some things out of the story.
At the very least, you need to get your license restored if you want to do any driving in the state. Your case won't solve that problem for you.