Many people who test positive never have symptoms but are still potentially infectious. Also possible that he is still in the pre-symptomatic stage.
"test doesn't work for him" is a non-starter, no reason any person is more likely to get false positive than anyone else. I've seen 99.9% specificity claimed for the UK test, in this case the probability of getting two false positives is (1-0.999)*(1-0.999) = 0.000001 i.e. 1 out of every 1million people who are tested twice will get two false positives.
Also, my understanding is that it is now a criminal offence to not self-isolate after testing positive?