How do you use an LED's log V/I curve to force an exponential current change in the LED for a linear voltage change?
LEDs are specified, as far as they are, to allow you to design power supplies to get a reasonable current through them. They are not intended to have any particular V/I relation, or be judged on their goodness of fit to any particular theoretical relation. This means if you want to use LEDs in your V/I converter, you are on your own to characterise them.
To give you a leg up, here are some curves I plotted for my own interest a few years ago. Sorry that the colours of the traces don't correspond to the colours of the LEDs. This plot was intended to be comparative, so I have not made a note of the temperature, and IIRC the illumination was typical lab lighting levels.
See how bad a 3 V zener diode is, compared to a LED. Notice the 1N4006 is a 'better' diode at both high and low currents than a 1N4148, so now you know which to use to put in shunt with your 200 mV input meter to protect it from accident.