You put a spectrum alalyzer inside and transmit to it.

This guy shows that a box of aluminum foil drops EM by at least 40 dBm. Chicken wire is more like 20.

https://youtu.be/9M6h1U9jgWs

A simple test would be to stick a few wireless devices in the box and check that they all loose signal once the box is closed. If you want to get fancy any place that does EM interference tests of consumer goods should have the room and equipment to test the box across a broad spectrum.

My intuition would be that the lid doesn't seal that good as it is. For long-term storage that's easily fixed by putting the shielding on the outside and gluing it shut with copper tape. Maybe you could instead add some copper-lined magnetic flaps that close the gaps.

The box has flaps lined with copper (https://www.instagram.com/p/B3Uwc9SHzot/?igshid=5dgayhtfayad). The box was a bit of a joke since I have no background in EE and couldn't find good reference material.

I think various electronics companies have test rooms for this and govements have actual EPM test rigs

This is not too healthy for the microwave

Put a glass of water in the microwave as dummy-load to prevent high VSWR from reflecting back into the magnetron. Standard procedure any time you're doing dumb shit with a kitchen microwave.

Good point, but it's probably the most accessible test device for high levels of RF flux. But based on quick wikipedia tour, just testing on the ~2.4ghz band of microwave oven would probably not be enough to verify the shielding.