Id personally ignore it meself.... and tell them to go forth etc...
I would say this is a bad plan. Its likely to lead to the addition of admin fees, passing of debt to debt collector, then potentially (depending on the debt collection agency) a CCJ and bailiffs, in that order.
When I've been charged for something I believe is unfair/unjustifiable I typically pay the amount and then fight to get it back. This then prevents the aforementioned harassment and other associated problems (damage to credit rating etc). Of course that only works if the payment of the amount in question and the costs to get it back doesn't put you in financial difficulties - I am fortunate in that it never has.
In this instance, given the (relatively large) fine and (relatively small) overstay, combined with the mitigating circumstance that the OP's wife was actually using the store, I would take the case to POPLA on "genuine pre-estimate of loss" grounds. Although something in the back of my mind makes me think a big case is going through the supreme court ATM on that very issue. Some googling would probably be useful.
In the second instance, I'd go to Aldi for an ex-gratia payment, given the amount you could probably get it back from them just by threatening to take them to the county court for it. Even if your reasons were spurious you could probably get it off them because the company cannot afford to defend a claim for £70. Their solicitors probably bill out at about twice that amount - per hour.
Just my 2p