Hey, the main thing is they recognize a bike is better than a car, that is progress.
Its an occasional thing, my official job description has the "...and other duties as assigned..." clause. No one gets mileage, except if the boss makes an exception. If anyone starts getting mileage I'm pretty sure I'd get it at the same rate. The previous boss took me aside and told me that if I ever didn't feel like biking just take a cab and she'd sign off on the expense report. I never took her up on it, even in the winter. Believe it or not I actually like car free living even in the bad weather. The career thing might be an issue. When I have to hang with the suits at HQ they chuckle when I show up in suit and tie and bike helmet. Funny thing is several have cornered me outside meetings with "My wife wants me to start biking, what do you recommend." Or "I bike on weekends, do you think I could take the family camping by bicycle?" things like that. Parking is such an issue I think most people see my biking to meetings or to the main job site as freeing up a parking space. This is especially true at the main site, if I started driving I'd bump someone from the assigned spaces.
On the mileage thing. Once I had to take a company trip to a part of New Jersery with uncertain public transport. Under the contract rental cars were disallowed but mileage at the government rate was allowed. I rented a car and charged the mileage rate- honestly. I figured I could eat a few dollars in the name of company loyalty. Well, I came out twenty dollars or so ahead. I told accounting that I'd accept just the rental cost but they said "We have to follow the rules we can only give you the per mile expense so you have to take the check for the full amount." So even if you own a car doing the rental thing for company business can be a good choice.