I have figured out how to do it and it appears to be somewhat undocumented in the ESP32-C6 Technical Reference Manual so I'm posting it here for reference.

It can be done like so, include:

#include "esp_system.h"
#include "soc/lp_aon_reg.h"

and add below lines:

REG_WRITE(LP_AON_SYS_CFG_REG, LP_AON_FORCE_DOWNLOAD_BOOT);
esp_restart();

It will boot directly to the ROM update mode.