It is extremely rare to have W2W carpeting not be color-fast. Now and then, when cleaning an older wool oriental rug, I can see some color showing up on the pads. Cleaning these rugs has never resulted in colors actually running, nor have I ever had whatever color appears on my pad transfer onto the lighter areas on the rug itself. But to enable your customer - and you - to feel comfortable, you could spray a little cleaner on an out-of-the-way area of the carpet in question. Then, using your stair tool and one of the small pads used with these, start to work on a small area. You will soon see if any color is coming off onto your pad.
And even if there is any color, you are highly unlikely to cause any actual running of the color/s in the carpet.