I think you'd be hard pressed to get the dealership to pay the charge. It is not a "fine" and you should not refer to it as such. It is a speculative invoice for an alleged breach of contract by the driver.
Horizon’s Notice to Keeper (NtK) is PoFA-compliant, meaning the keeper can be held liable if the driver isn’t named. If the signage at the Tesco car park clearly restricts parking to Tesco customers and conforms to the BPA Code of Practice, then the driver is likely contractually liable for breaching those terms.
However, the dealership’s advice to park there introduces a mitigating factor. While it doesn’t override the signage, it may support a challenge based on fairness or inducement. You should appeal to Horizon without naming the driver, and if rejected, escalate to POPLA. Also consider pressing the dealership to cover the charge, especially if their advice directly led to the breach.
