Relays clicking is a good sign. Is the monitor an LCD or CRT monitor? Sometimes LCD monitors are extremely sensitive, especially when a video mode is at the limits that they support. The pass through wouldn’t change the signal too much (possibly just voltage drop across the relays) but with some LCDs...
Can you try with a different VGA cable? Maybe switch the cable between the Picasso output and the monitor and the cable between the A3000 VGA and the Picasso input.
Also try a different monitor.
And, which connector on the Picasso are you plugging the signal from the A3000 in?
The lower VGA connector on the Picasso (in the case of the A3000 the one that is closer to the center of the case) is the input and the upper VGA connector is the output.