How to measure small AC currents from mains?
You can loop a thin wire many times through the hole in the core to increase the sensitivity.
For example, you could use this current transmitter from the same supplier with a 10x loop and get 100mA full scale, which would just accommodate a 10W bulb on 230/115VAC mains (about 40 or 80mA).
An alternative would be to purchase a handheld DMM that has an isolated serial interface, but that would require batteries.