Actually I've read claims that ESP32 C6s are pretty decent battery-consumption-wise. So much so that I bought a few, hoping to make at least a doorbell out of it. Alas I don't have a device to measure microampers, so I guess I'll just see how long they're fare..
You can use ohms law - let it draw power through a 10k resistor, and put your multimeter across the resistor. Every .01 volt is 1 uA. This also means that if you're powering it with 3.3v and it browns out at 3.0v, you'll only be able to draw 30uA before browning out.
You can use a different resistor according to the power draw and how sensitive your volt meter is.
You'll probably need to power it up with the resistor shorted, and only remove the short once it's in sleep mode, to measure the current.