Hi,
you are looking for the "best", but you don´t say in which regard it should be the "best".
There is no one and only "best" current source. (otherwise there was no reason to build different ones)
Often the problem is that that one did not decide the requirements ... or the target.
Deciding the requirements should be the first thing when building a device.
Currently we just know about a moving metal ball and the resistance range.
But we don´t know what make the resistance change, nor the environment. We don´t know whether you are interested in amplitude, phase angle, frequency, .... accuracy, precision ... how often you want to measure.
We don´t know whether you want a LED to become darker or brighter, or you want to use an analog measurement device, or maybe a microcontroller.
It could be simply switching a relay, or you could be a complex mathematical signal analysis.
The metal ball could be moving 1D, 2D, 3D .. or rotate in several ways...
****
Back to the current source: (from simple to more precise)
* It could be a simple resistor, mabe a PTC or NTC ... and thelinearity is good enough.
* One could by a current source IC.
* One could use a resistor transitor circuit,
* ...maybe add a zener.
* one could use a voltage regulator
* or an OPAMP circuit
* or an advanced OPAMP circuit with precision resistors and precision voltage reference.
We also don´t know about the power source. Is it mains powered, or do you need a low power battery operated device.
Klaus