I thought I answered this one already
Get a Boss FV-200 Keyboard Volume pedal (or equiv.) and ask the tech department to rewire it for you. There's two inputs and two outputs.
Plug your guitar in one of the inputs (or split it to both) then take output 1 to your first amp, and output 2 to your second amp.
With the pedal in the fully back position you'll get just amp #1 then as you rock the pedal forward it drops the volume of amp #1 and brings up the volume of amp #2.

If nobody knows how to rewire it try reminding em that it was used by guitar synth players so they could fade the guitar sound in.

Better yet, somebody probably already invented a pedal to do this anyways.

Alternatively, you could just have your sound guy twidle the balance control.