It's your money and as long as you don't mind being marked as a "mug" for future scamming, by all means try and pay them. Don't think for a minute that you are dealing with a company that cares about your frustration of contract. They are a firm of ex-clamper thugs whose only interest is how much of your money that can scam out of you.
They have been labeled "cowboys" and "scammers" in parliament. When the upcoming "Private Parking Code of Practice" Act becomes law, hopefully later this year or early next year, they won't be allowed to rip off victims in circumstances like yours where you were broken down. However, for now, your best option is to defend any claim should they decide to pursue you for the alleged debt.
As you have already experienced, they don't make it easy for you to pay the
bribe discount, charge. They have now upped it to £100. You can decide that you don't mind paying them or you can fight it. Your choice.
If this ever got to court, your defence would include the fact that you were no "parked". You were broken down which is a vicissitude that has been used as a defence on many cases. The appeal court decision in
Jopson v Homeguard [2016] B9GF0A9E, is the persuasive argument that would be used.
One question though... why did you disappear for 15 minutes? Were you aware that you were on private land and that you may be liable for a charge? Was the NtD on your windscreen PoFA compliant and did the follow up NtK arrive within the prescribed time and with the necessary wording to hold the keeper liable? The driver and the keeper are two separate legal entities in private parking cases.