The Chicago Manual of Style says you should have "Grandmother", "Dad", and "Aunt Doris" but "my aunt Doris" - it says:
kinship names are lowercased unless they immediately precede a personal name or are used alone, in place of a personal name. Used in apposition, however, such names are lowercased.
In a case such as "my aunt Doris", "my aunt" is considered to be in apposition with "Doris", just like in a phrase such as "my favourite country Spain". (Commas are probably optional.)
You would also lowercase "I met his father", "a boy's father is the most important person in his life", etc. But use upper case in "I shot Father" or "I love you, Father".
In the examples given, you would capitalise "Big Sis" because it is used in place of a personal name in "Big Sis? No. She gets another point." and immediately precedes a personal name not in apposition (it is used as kind of a title) in "Big Sis Tia? No. She gets another point."
Having said that, this is only the Chicago Manual of Style, and other places may do it differently.
Source: Aunt-ing and uncle-ing, Grammarphobia, November 19, 2014