The LM2596-based module I bought is, of course, a fake. Apart from the dodgy laser-etched case markings, the switching frequency is only 55kHz instead of 150kHz. So now I have to find a real LM2596 and transplant it onto the board, or else risk the fake part not quite being reliable enough.

In the meantime, I've increased the main PSU voltage by a couple of 100mV, and now the RPi is quite happy and doesn't have as many low-voltage warnings.

Maybe I'll park the 'upgrade' idea for a while and see if I still need it...