Circuit to control an RGB LED strip using potentiometer sliders to change colors
Figure 1. The RGB LED controller.

Potentiometers are designed to handle small signals at low currents. In the device you referenced the potentiometers are used to control a transistorised circuit which will handle the relatively high currents drawn by the LEDs. Dimming will be achieved using pulse-width modulation (PWM). This switches the LEDs on and off at a rate faster than the eye can detect and by varying the on-time pulse-width dimming can be achieved.



Figure 2. PWM signal transitioning from high pulse width (75%) to low (25%) and back again. Note amplitude remains constant. This will result in an apparent brightness of 75% and 25% of full brightness. Source: LEDnique.

The simplest solution to this will be to purchase the unit, open it, figure out what resistance value the potentiometers are, disconnect them and wire in sliders of the same resistance value having identified the pins on each.

Your biggest problem will be making a nice panel for the sliders.