Sqlite's limitations have nothing to do with PHP. But you're correct that it's not intended for use as a multi user database. The limitation is in concurrent writes. If your app is read-oriented, then there is not a problem.
However, I'd personally always use postgres in this case, myself.
This article has the typical dichotomy of a piece discussing the medical aspect of addiction: after waxing on for paragraphs about the medical nature of addiction, and how the rats she studies are helpless to avoid relapse, it then ends on a moral note, stating that "the choice [to maintain sobriety] is his".
Well, if addiction is in fact a medical problem, then the choice is not entirely his, is it?
For people like the father, I'd strongly recommend getting on and staying on methadone. It's not easy, but it's a real of a lot better than going to prison and destroying one's family.
However, I'd personally always use postgres in this case, myself.