Should I show my father pictures of my sister with the young man to prove her relationship, while taking into account not losing my relationship with her or upsetting my father?

If what you mentioned about your sister's relationship with a young man and her going out with him is true, then she is involved in a reprehensible act and a trial (fitna), and the trial of men by women is among the greatest trials. You did well by advising her and alerting your father, as he is responsible for her and must fulfill his duty towards her, in accordance with Allah's saying: "Allah instructs you concerning your children" and His saying: "O you who have believed, protect yourselves and your families from a Fire." Your father should have investigated and reformed instead of denying. You can confront your sister with evidence that proves the validity of your accusations, and threaten her with showing it to your father to deter her, while avoiding showing the picture if you fear a greater harm. Continue to advise her and supplicate for her repentance and righteousness. If you obtained the pictures by spying on her phone without her knowledge, then this is forbidden.