After testing if the ESP32-C3 could wake from a rising edge GPIO, it does not seem to work. The datasheet was a bit ambiguous, but after writing the RTC_CNTL_GPIO_WAKEUP_REG for edge trigger the MCU would not wake when a signal was applied to the GPIO pin.

void RTC_DATA_ATTR wake_stub_sleep_counter(void)
{
    wakeup_cause = esp_wake_stub_get_wakeup_cause();
    if(wakeup_cause & 0x04)
    {
        // Wakeup cause GPIO
        ++wake_count;
        intr_type = intr_type == GPIO_INTR_HIGH_LEVEL ? GPIO_INTR_LOW_LEVEL : GPIO_INTR_HIGH_LEVEL;
        gpio_ll_deepsleep_wakeup_enable(&GPIO, GPIO_NUM_4, intr_type);
        // Go back to sleep
        esp_wake_stub_sleep(&wake_stub_sleep_counter);
    }
    else
    {
        esp_default_wake_deep_sleep();
        // Return from the wake stub function to continue
        // booting the firmware.
        return;
    }

} 

I measured the time taken for the ESP32-C3 to wake and return to sleep as 12.4ms, using a 1 ohm resistor inline with the power supply, showing the current draw in this mode to be around 8.4mA. Assuming an average of 1 pulse counted per second, we can estimate the battery  life running from a 800mA 1.2V AAA not counting the wakeups to connect to WiFi.

0.8*1.2=0.96Wh
3.3*8.4e-3=0.0277W
0.97/0.277=34.63H (if on continuously)
34.6/(12.4e-3*2)=1395Hours (if one pulse per second on average)