Well, I'd take the carrot. Home to duty milage is a complete anachronism and how it has survived so long I'll never know. It is only worth washers to anyone who lives within a few miles of work and why should the company subsidise those who choose to live a long way away?
Anyone remember the good old days when we actually had to claim it shift-by-shift? 4p a mile - same rate as now only with additional admin costs...