What action you choose to take is your choice (as it's your time and money at stake), but it essentially depends on your attitude to risk, and how much you value your time and effort vs your willingness to fight them.
There is an argument to be made that by failing to properly specify the period of parking, they have not complied with the requirements of Schedule 4 of the Protection of Freedoms Act (2012), and are therefore unable to recover the charges from you the registered keeper. CPM are a member of the IPC, which means that neither they nor the 'Independent Appeals Service' are likely to accept any appeal, regardless of merit, so this argument would need to be run in the County Court (small claims), assuming they went that far.
There may be other arguments based on the quality of the signage at the site*, but we'd need to see images of the signage to advise on that.
*If the signs say 'Permit Holders Only' or the like, there's an argument to be made that no contract was formed with the driver.
We cannot guarantee success, but if you're up for a fight, we can advise.