None of that is logical at all... well AFAICT.
First there are 2 common ways to run leds. Constant current and constant voltage.
Initially I assumed this is just your standard constant voltage array based on 24v and of course diodes and resistors.
The timer is a simple mosfet switching circuit that chopped the 24v up into itty bitty pieces..

So the dimmer is composed of a small logic circuit, a voltage regulator to power it, and a mosfet as the PWM switch.
Easy peasy..
Just remove the dimmer, use an "appropriate" MOSFET and circuit hooked to the pi..
Also pretty easy peesy.
Well first one needs to confirm that the specs of the mosfet one picks fits what is needed, like fully open at 3.3V (or 5V, whichever is used) and can handle 24V and the amps at 100% duty cycle.
BUT then there is the 2 "mysteries".
1) Why does the controller only dim to 10% and why did you measure the voltage at said 10% to be the AVERAGE of 13.5V.
2) Why would they list the "real" wattage at 25W.
Makes no sense to a constant voltage array that would measure the entire circuit
voltage across the 6 leds and the resistor which would be 24V. The current for a 3w led would probably not be
8.3W per string, 3 strings.
348mA
That would "fit" 1W class leds . Average v(f) of 3.6
3.8 actually works better.
Did they lie stating "3w" leds?
Actually this could matter if you decide to swap leds.
One other semi-oddity for the series. For 18 diodes and assuming 24v the usual string is 5-6 leds. Would seem to fit except for the fact that it uses "violets" which generally have a higher V(f) than rb and blue leds.
Some of the larger units do not divide by 6 but the ps's are???
3.2-3.4v for blue/rb and 3.5-4v for violets. So worst case 4v x 6 = 24V So close enough I guess considering the strings aren't 100% violet.
So though I still believe this is a simple cv array there are some questions that need answering.
As such a simple attachment to a pi
Now the oddly complicated one:
There is a small possibility that the led board has "proper" constant current drivers on the
led board itself.
A 24V CV switching power supply to feed them.
3 with a current set point of say 350mA, no resistors of course.
Or one at 350x 3 (1050mA) but "shared" over 3 parallel strings.
Now that does not usually use an in line dimmer module to chop the input to the drivers though it is a possibility.
Either way the answer is the same. Low side mosfet control
And no.. you cannot change the voltage and have things work well or properly AFAICT.
It either wont work or is completely unnecessary..
You can dim leds in a constant voltage array by using a power supply adjustment but that usually falls in the range of 0 .1 v to maybe 1-2V
You may get away with bit if they happen to use boost drivers but THAT is HIGHLY unlikely. It would be the first I know of.
Try hooking a 12v ps to it with the dimmer removed.. Probably won't light or be real dim.
This is just my way of looking at it.. Corrections are encouraged if anything is illogical.
After proofing this a bit it dawned on me that there is one more remote possibility.
That the dimmer actually contains a "proper" driver.
But it would need to contain not just the driver circuitry but also the full logic circuits to control said driver.
It has been done before though.
BUT, and correct me if I am wrong, light works fine without the dimmer.
Leading back to drivers on board or a simple cv array.
ONE last thing.. If that little dimming box was ACTUALLY dropping the ps voltage from 24v to 13.5V wouldn't that generate a lot of heat?
Equiv to a 30 Ohm resistor (10.5V 350mA), 3.6 watts?
.