GFCI's are delicate They do go bad. Mostly in outdoor locations. It is useful to determine if your smoker is tripping it or is it a neusant trip in the GFCI it's self. We need to keep in mind what a GFCI does. All it does is constantly monitor the circuit for ground faults. They are when voltage(amperage) leaks to the cabinet or chassis of the smoker. It monitors and trips at very low points. 0.006 ma to be exact that is not much.
OK we got that behind us now on to the fun part. The test. You can do this with only your multimeter.
Unplug the cord and look at the plug end. You will see 3 prongs. They are all different. One is round that’s your ground the larger flat one is the neutral and the other is the hot. All you have to do is turn on your meter and set it for resistance. It might be marked by a upside down horseshoe symbol. Put one probe on the ground(round prong) and one on one of the others. If your needle moves it will probably be the smoker. If not look at the GFCI it's self. You can do the same test on the other prong and then a new test since you might have a bad ground wire in the smoker. Put one lead on the smoker box and one one the hot lead. If you still have no movement in the meters needle it's a good bet you problem is in the GFCI. The tests you have just done is to tell you if the smoker and the power circuits are making contact some wear.
If you have a ground and the needle moves you need to check the connections and both ends of the ground wire. Clean the ground rings and the area it connects to. On the MES 30 some times the end of the ground wire is away from the element(far right side) it gets grease under it and grounds the smoker. .This is not a big deal you just need to drill out the 20 something rivets that hold the back on. When I reinstalled the back of mine I used 3/8 x ½ inch truss head screws. They were stainless steel with a Phillips type head. That makes screwing them in easier and the truss head is bigger than normal. It's like having a built in washer in the screw head. They worked great. I used a 1/8 Bit to drill out the rivets I needed 2 drill bits. I would not do this except as a last resort.
Another thing to think about if you have not found anything else is that the element may have sucked up some moisture. Some times it just needs to be run on a non GFCI receptacle and it will dry itself out. If need be check out my post on trouble shooting elements. Jted