and was looking for a way to pay/avoid a parking charge. Unfortunately, the screens to register number plates are located in store, so there was method for me to register or pay for my stay.
One of the immediate hurdles with this argument would likely be the length of stay. You'd need to provide a credible explanation as to why it took 22 minutes to park up, read the terms and conditions on the signage, realise the store was closed and then leave.
There's an argument to be made that no contractual offer is communicated to those who are not Aldi customers. Also a potential argument that the entrance signage was not prominent depending on which way the vehicle entered (the strength of such an argument would probably depend on the quality of the signage as a whole).
Unfortunately, it was sent to another registered driver on the vehicle who has only recently notified me.
Bear in mind that if the notice is addressed to someone else, it is them who must respond. Either by appealing (if they can) or by naming you as the driver.