Ooh this was my suggestion! For the record, I have a relative with this name/nickname set up. She doesn’t go by it, but her family nickname is Tate. Totally works in that setting.
I think Katherine “Tate” over just Tate. Alongside Scarlett, I really think Tate would prefer a more feminine and formal option.
I agree with others, you do run the risk of people calling her/mishearing Kate. Maybe using Katherine gives permission for other less preferred nicknames (Kate, Katie, Kat). I think it could be easily corrected, but would require some assertiveness.