Drive your car in on Monday. Bring your bike, clothes for the week, etc. Ride your bike home and then to/from work til Friday. The car stays at work during the week. That would be one way to make it work and to commute by bike 8 times a week.
Keep pushing for a company car.
My commute recently went from about 7 miles to almost 25 miles each way. The building where I now reluctantly work is quite a distance from our headquarters, and we have a lot of meetings there & in DC. If we are lucky, we can use a GOV. Most times it's not available (it's shared w/other offices) or just not practical. So I drive my own car and charge back my mileage to the poor taxpayer. Forget about commuting by bike, as the new building is located in the worst of suburban sprawl (major, unsafe highways) or even commuting by public transportation. Luckily, this situation is not forever---I'm free to find another assignment next September. I love my job & my boss, but the commute is stealing between 45-60 additional minutes of my life every day.