I am a mom of four boys ages 16,8,7 and 10 months! My dad is sick with cirrhosis of the liver and pancreatic cancer, although he is stable. When he does have to go to urgent care my sister who has no kids dandse to pack up the kids and take them with me to my dad's appointments because she says it's not fair that she goes and I don't! I'm so busy with my kids and husband, I never get a break for myself. I bring my dad four days out of the week to my house to care for him. On the weekends my sister goes clubbing to nightclubs with her friends. While I do laundry and catch up on chores!
https://www.agingcare.com/questions/Should-I-be-taking-my-kids-to-my-sick-dads-DR-appointments-185660.htm
As for you having to bring all the kids with you for your Dad's doctor appointments, see if your sister could babysit since he is tired of going to his appointments [many of us can understand that, like if we see one more waiting room we will scream].
Just because your sister has no children [which was her choice] it doesn't mean she doesn't have a house to clean or groceries to buy. Yes, you have more cleaning and more groceries, but that was your choice.
See if you and your sister can work as a team, to help each other out more with your Dad's care. If it becomes too much, maybe it is time to get Dad another level of care, like a paid caregiver to come to his house.