An op-amp constant current source like Zero999 mentioned may be a good way to go. With only 200-300mV of headroom you'll want to use a FET instead of a BJT as the pass transistor and that means compensating the loop for the gate capacitance. 200mV at 50mA means max 4R for the sense resistor, transistor, etc, so the FET will need to be a fairly low resistance relative to the actual power handling ability you need and gate voltage you'll be able to apply, so a bit of care will be needed in selecting it. In reality you might have more headroom than that, but it partly depends on how the module is built. You'll need a a reference for the drive, of course, which can be divided from a regulated rail, or if your MCU has a built-in bandgap reference that you can bring out that could work. Even better if you have a DAC you aren't using, then you can even make the backlight current programmable.