Your employer is correct and no they should not be paying anything. It seems you have a choice - use approved methods of transport which they will re-imburse you for or use the car at you own expense and by the way at your own risk.
An indemnity and simple insurance issue arises. In order to pay you to use your car for business purposes, the company would normally require you to indemnify them against any accidents, claims or losses. In order to do that you would need to declare to your insurers that you are using you car for business purposes. If you haven't done that then business trips in you car means you are driving uninsured.
I would guess your employer isn't interested in all that palaver so they don't want you using your car and won't pay you for it.