A bad girlfriend?

My bf and I live together & we work 2 blocks away from his moms house. Every Monday, Tuesday & Friday, he has to take his 12 year old sister to school around 8am. We car pool to work so I’m always with him whenever we clock out of work (around 7:10-7:20) his mom is always home when we arrive there around 730am and just leaving for work & we always wait until 8am to drop her off (she won’t leave earlier because she says she’s busy doing her hair. we have dropped her off earlier but she doesn’t get ready early anymore) I’ve been going with him for about over a month now & the last time We went there around 710am, I was so exhausted I feel asleep cold on the couch. Yesterday, I decided to tell my boyfriend if it was ok for him to drop me off home first and then take his sister to school? we only live 1 1/2 mile away and her school is down the street from his moms house. He said no and that I have to deal with going with him to take her to school or just take my own car to work. I told him that I pay most of the rent and bills and I can’t afford to take my car to work 3 times a day (we live together so we save money car pooling) so I politely asked if he didn’t mind giving me gas money so that we can go to work in separate cars and he can go ahead and take his sister to school. He got furious and yelled and said for me to pay and why he should pay (again, I pay most of the bills at our house so the gas would help me a lot) I finally told him, since his mother leave to work around 730-735am and school is down the street, why not have her take her daughter to school some days to help out so we don’t go 3 times a week? He said he and that he wanted to be the one to take her to school. I’m annoyed and upset & idk what to do. I don’t mind him wanting to spend time and take her to school but I just want to go home after working a hard day at work!. We work over night 10am-7am. What should I do?