At least in infectious disease epidemiology as a field, "Asymptomatic COVID-19" is a perfectly accepted term. SARS-CoV-2 "referring to the virus" is for things

At least in infectious disease epidemiology as a field, "Asymptomatic COVID-19" is a perfectly accepted term. SARS-CoV-2 "referring to the virus" is for things like "SARS-CoV-2 has a number of receptor binding sites..." If you're talking about what a person has, even if it's mild or largely symptom free, we use COVID-19.