Do you receive a cash allowance for a car?
Is it in your contract that you need access to a car for work?
What mileage rate do you receive?
If you receive cash allowance for your car, or your contract for the job states that you need to have access to a car then i don't think you have any grounds. Is there a company car schemen you could opt into? Depending on the BIK it could be worth considering.....or perhaps a pool car system?
Perhaps you could persuade your employer to pay your underpaid mileage through petty cash asap, or could they advance you some money to purchase tax/insurance and you pay back over a certain time period? (not sure what tax implications are for this).