How to amplify the resistivity range of a rheostat
Since we can't know what the black box actually does with the potentiometer – does it really just control the amount of current flowing through the leads, which is used internally, or is the voltage used, or is, in fact, the potentiometer part of a feedback loop that limits the turn-on time of your inverter's switch mode power supply, there's only one thing that certainly works:

Forming a larger resistance. I'll be frank and say that with a big margin, the easiest solution would be to simply buy a 5 kΩ potentiometer and use it instead what your dimmer switch has.

There's tricks how you could implement this e.g. using transconductance amplifiers, but if it's really the case that the original potentiometer is part of a feedback loop, you might get into trouble for changing the phase of signals.