If you want a fixed constant-current sink, you could try this circuit:
Diagram:
I had to check in LTSpice, it looks like it works fine but needs a 12v supply to bias the transistors.
The example shows it sinking 0.5A from 4.2v down to below 1v.
The MOSFET would dissipate around 2W, so it needs a heatsink.
Yours,
TonyWilk