

If you’re designing for production then talk with the assembly guys, they will have options and cost estimates.
If we’re talking a one of, then just try. It’s only 500mm so not that much can happen. Especially at the slow speeds you mentioned. 20kHz isn’t that fast compared to the +100MHz that a cat5e is designed for when using differential signalling.
Are plugs a must? If not I’d consider soldering the wires. At least for testing the cable’s suitability.
If you must use plugs, then don’t use rj45. Some yahoo, like yours truly, will come along and go “neat, an undocumented ethernet capability” and fry something expensive. Have a look around DigiKey or where you get your parts, and find a less common plug.
And if you end up needing more wires, then get an old parallel ATA HDD cable from the early 2000s. They have 80 wires and are free at the dump.

I suppose you don’t have the original power source, if you did then measuring the output would be my starting point.
If it’s LED then it’ll probably be a single LED using PWM. In that case try figuring out where the switching circuit is.
If it’s not LED, you should measure resistance between the wires, that would tell you which one is the common. .