sounds like you're using the deicer on a line that can't handle the extra load. Maybe use an extension cord and plug it into a receptacle that is underused/capable? You'll prob need a heavy duty extension cord, 14 ga minimum.
Or, you're using a gfci that is rated too low for your use. If the gfci is rated to blow at 10 amps and you're pulling more, well, that's going to happen. A typical receptacle should handle 15 (before the fuse or breaker blows). That said, it is possible your deicer is malfunctioning and pulling more than it should. See if you can check the specs. If it shows less than your gfci cut off, then I'd suspect your deicer is the problem because it shouldn't blow. Also, any moisture around the gfci will also cause it to reset. If you try the extension cord and all is well, suspect the gfci.
Generally speaking, the deicer is just a heat element. Electricity flows, heat is created, voila, job done. When the deicer doesn't work, there's no heat. So, my bet is if you circumvent the gfci, you'll get the deicer to run properly.
hmm, just re-read your post; 3 different gfci, huh? Could still be the deicer is drawing too much relative to the cut-off. Try circumventing and see if you lose a house fuse/circuit breaker. If so, definitely the unit(s) aren't compatible. I still think it's the gfci limiting that is fouling up your system, though.