Travel to and from work is included in my 8 hr work day could I get a shift differential?

I am a K9 Handler. My shift is 4am-12pm and they include my travel to and from work into my 8 hour work day. So basically I get 2 hrs travel and work 6hrs on site. I have to be at work by 5am and was told if I get there before 5am I don't have to wait to work since I'm already on the clock. I use my own vehicle to transport my K9 partner. They do not have K9 vehicles! I don't get paid mileage or a gas allowance or any kind of maintenance for my vehicle. I was told that I must only travel to and from work while I have the K9 in my vehicle. According to HR policy if a shift begins before 04:59 your entitled to the shift differential. Some say I should get it because my shift begins technically at 4am, others say I wouldn't get it because traveling doesn't count even though its part of my 8 hour work day. I also work this shift M-F. Do you think am I eligible for the shift differential?