Usually to check current with a multimeter you have a separate terminal to attach one of the leads for current measurement (sometimes two) but there is usually a limit, often 10A (on my Amprobe, for example). You have to connect the positive wire inline so that current flows through the multimeter. This won't work for a heated bed which may often exceed 10A at 12V. I'm not sure whether you can get a good non-contact measurement at 12V with a clamping current probe (which goes around the wire(s) and measures induction) but that's usually how higher current flows are measured.
I'm also not an EE or electrician so take my comments with a grain of salt...
You can also measure the heat on the MOSFET heat sink using a thermocouple - if it's heating up when you are trying to heat the bed it's working, if not it may have died. I'm not sure how failure prone those are.
I had problems initially using an out-of-spec power supply which was supposed to provide 20A on the +12V rails but didn't. Usually power supplies either work or something blows and they stop working, and you would smell something melted in the power supply - they don't usually have separate protection for both of the +12V rails. But you eliminated that by swapping the controller board, so it's either the connection on the screw-down terminal block on the controller board (I'm assuming you tinned the wires and you're using heavy-gauge stranded wire for the heated bed) or the MOSFET on the non-working board has died.
