BA-DUM! I have an idea!! The 10K pot controls the pulse width of an oscillator that in turn turns the LED on and off. Either dark, or full brightness, nothing in between. This light shines on the LDR which has a capacitor in parallel to it to smooth out the PWM signal. You'd want a small capacitor to still be able to achieve fast transients of movement. TAU [the greek letter] =R*C, R is 1M Ohm at max so you go from there and calculate the minimum frequency of the oscillator to achieve a full swing of the movement extremes in a given time.
Let's assume you move the stick left-right-left-right-.... 10 times a second. You nee approximately 5 TAU to fully charge or discharge the capacitor. 5*TAU=1/10 = 0.1 seconds -> TAU=0.02 seconds. C=TAU/R = 0.02µF= 20 nF. According to Nyquist's theorem, we need to sample at least twice the amount of the frequency we want to measure, so T=0.02/2=0.01 sec. With the frequency f=1/T we get 100 Hz for the oscillator's frequency.
Now here comes the electrically easy part
You could just have the LED shine through a piece of plastic with varying opacity fixed to the stick. As the stick gets moved around, so does the plastic piece, resulting in different amounts of light on the LDR. Such a piece of plastic can be made by printing a dithering on it. That thing needs NO CLEANING EVER but is probably not that easy to build.