Hi,
I think you have to swallow the bitter pill, and use one preset per chord. But if you use an external midicontroller, you can directly switch to all presets via bankselect and programchange messages. Additionally you can use controlchange messages to switch single effectblocks in the signalchain like overdrive, chorus or delay on and off as desired to alter your sound. And if you use a parallel signalchain in your setup, you can blend/mix two amps via midi using the cc-messages for the exp-Pedals to control the different ampvolumes.
So all in all you can store 512 different chords for your JTV an use two different amps with various pedaleffects switched on or off. And if you don't need 512 chords for one song, you can store as much chords per song with the sounds needed as there is free memory space available. :-)
Have a look at the Behringer FCB 1010, wich is a very versatile Midi Footcontoller, to realize your project. I hope this will help you, and perhaps this will help others, when you tell us, if this works, and how you finally setup your gear. :-)
best regards
Alex