How do I charge this odd 4S Li-ion circuit?
Well most big companies have battery charging chips, like
https://www.analog.com/en/parametricsearch/11471#/
Linear Tech also used to have equalizer chips. My RC-Battery charger has one.
In general... a µC, and a powerfull buck(boost) Stage can charge Lipo's that's how my RC-Battery charger works. You just need also to measure voltage and current as lipos need tight control. If you do it wrong, prepare to call the fire department.