It's not a fine. It's an invoice.
OP should 100% not pay it yet. First port of call should be to get Sainsbury's to cancel it. Second port of call should be to appeal it.
Ultimately, I think OP would be well advised to not pay at all and let them threaten her. Most of these are dropped when the car park company encounters someone who won't back down, and those that aren't are very often lost because the car parking company hasn't complied with their own obligations.
Also, as long as you don't ignore a court order to pay something, you won't end up with a CCJ. That is scaremongering nonsense.
No one should ignore it, but they also shouldn't just jump straight to paying it.
OP - how big is the car park? The new code of practice allows 5 minutes consideration period for smaller car parks and 10 minutes for larger car parks; plus ten minutes possibly for a grace period depending on the terms and conditions. So you might actually have been within the allowed grace periods, or just barely outside them. There's also a good chance the signage in the car park is not adequate.
Other point - at no point should you ever tell the car parking company who was driving. Their obligations are different if they can't prove who was driving and if they have to hold the keeper liable instead, and they often fail to meet those obligations.