That's why I like to work with my GT-10 and my DAW at the same time since you can have 2 channels at the same time; I have my distortion in channel right and the unprocessed signal in channel left (you can have the raw signal coming out from the GT-10). That way, I play with direct monitoring and mute the software monitoring, so I hear the GT-10 without lag and record 2 tracks at the same time; one is what I just heard, the other is the clean signal.
With the same setting but the direct monitoring turned off and the software monitoring ON, I have another option for a live use: automate everymotherfuckingthing.
I set up a good clean sound rather than an unprocessed one, and keep my distortion on the other channel. I can program things like "Mute clean on bar 17/Unmute Distortion of bar 17" and vice versa. Have my delay stop at 75% of bar 24, the value of the time delay change, put a flanger on the last note of another bar... have my volume fade out here and there, solo boost when needed.. you get the picture
If you plan on using the GT-10 with an amp, use the GT-10 as the pre-amp and the amp as a simple power-amp, you do that by using the "4 cable method".