Sherman50, I think you are making this too hard unless you are trying to shave a few minutes off your start time. Why not just leave the hotel and go get some gas with a receipt?
I can only speak for personal experience. I am a frequent traveler (120+ nights a year in the Marriott chain) as I travel for work and play. I never check out at the front desk as they always just leave the receipt under the door overnight.
I just looked at my latest receipt from finishing my 48/10 and I do see a space for a date AND time for check in and check out on the hotel bill. I did notice that the departure time was blank since it was the receipt that they put under my door overnight.
I would confirm that the hotel can provide a receipt with the departure time. What also might work is arraigning to get a credit card receipt when you leave at the front desk.
Although we much prefer the start and end receipts be from a gas station, we will accept any computer-generated receipt showing date, time, and location. That will be your official start time and location.