Figure 1. The CIE colourspace with a "best-fit" line running through the whites.
To do what you want would require some good fortune with availability of the right LEDs. To be able to adjust along a best-fit spectrum of whites requires that you have two light sources with colours on the outer ends of the line overlaid on the graphic of Figure 1. Colour control in this case would be fairly simple: blend between blue only to gold only.
If the colours don't line up as desired then you would need at least three colours.
Figure 2. With three colours (located at the points of the triangle) any colour within the triangle can be generated.
In this case control becomes a little more complicated but could still be done without resort to a micro-controller.
Figure 3. A green-blue pair and a green-red pair could generate the two colours at the ends of the grey line.
If a green-blue set of lights is adjusted to generate the blue at the left end of the grey line and a green-red pair is adjusted to provide the hue at the right end of the line then, by varying the relative intensity of the pairs we could travel in a straight line through daylight, cool white and warm white.
simulate this circuit – Schematic created using CircuitLab
Figure 4. A very simple 0 - 10 V dimmer control.
How it works:
- This control is designed to work with 0 - 10 V LED controllers.
- It uses two 2-gang linear potentiometers (pots).
- R1 controls the intensity. As the potentiometer is turned from minimum to maximum the wipers' voltage will increase from 0 to 10 V.
- R2 controls the temperature. Note that the potentiometers' inputs and grounds are reversed. When turned all the way to the left the cool control will get the full value of the intensity setting while the warm control will be grounded and get zero volts. When turned all the way to the right the situation will be reversed.
- With R2 in the centre position both lights will be at 50% intensity. This is intended design to maintain intensity across the range of the temperature control pot.
- The 1 kΩ and 10 kΩ values are chosen so that R2 doesn't load R1 too much (as this would distort the desired linear adjustment.
Figure 5. The potentiometer connections on the PWM controller are not grounded.
Unfortunately your Amazon controller is not suitable for this scheme as the pots have neither end (points 1 and 3) grounded (to 4) and so must not be using a control voltage as required.
Many LED controllers are available with 0 - 10 V control inputs. While it may not suit your application, my article Dimmable mains PSU control explains some of the control features.