I don't think a GFCI is going to work with the 3-line power cord. You should probably look at the 3-in, 4-out spa panel GFCI solution, and plug that into your dryer outlet, and get the 4-wire cord from High Gravity.
The spa panel jeff is talking about doesn't have to be a permanent install. It can have a short cord with a plug that plugs into the dryer receptacle when you need it. It takes a 3-wire 240vac input and converts it to 4-wire 240vac output with gfci protection. It isn't as ideal but it gets the job done and increases safety. At least that is what the experts tell me.
*Everything I say is fully substantiated by my own opinion