Is it possible for me to indirectly give hsv-1 to someone? 5’10” 193lbs man, 25 years old.

I’ve been dating my girlfriend for a little over a year. I got std tested recently and she did too. I am positive for hsv-1 and negative for everything else, she is negative for everything. This was my first time getting std tested ever in my life so I had no clue I had it, I’ve never had any symptoms at all, I’ve never had anything on my lips or any herpes symptoms. Her little sister seems to have herpes, she did not get tested but the doctor looked at her lips and said that she has that since she’s been having flares. My girlfriend shares lipsticks with her little sister all the time. My girlfriend’s sisters are blaming my girlfriend for it since she’s been dating me and I have it and she told them everything. I talked to a doctor and even recorded her and she mentioned that there is no way for my girlfriend to transmit the virus to her sisters since my girlfriend is negative for hsv-1, she also said that for you to infect someone you need to be showing some sort of symptoms. This is getting heated a lot in her household since the little sister keeps having flares and my girlfriend is having to pay for the expensive medication, my girlfriend feels really guilty since she does share lipsticks with her little sister. Is the doctor correct in her assessment or could I have possibly cause this issue?