You can't. You have a single writer - it's one of the many reasons sqlite is terrible for serious work.
You'll need a multiprocessing Queue and a writer that picks off sentences one by one and commits it.
What do you consider "serious" work? We've served a SaaS product from SQLite (roughly 300-500 queries per second at peak) for several years without much pain. Plus, it's not like PG and MySQL are pain-free, either - they all have their quirks.
I mean it's not if he's got lock contention from BUSY signals, now is it, as he implies. Much of his issues will stem from transactions blocking each other; maybe they are long-lived, maybe they are not. And those 3-500 queries --- are they writes or reads? Because reads is not a problem.