The pic with the AD8603 is probably the way to go. The system would probably have to operate on plus and minus 2.5V power supply. Operating near ground is tough without what's called split supplies. +-2.5 is not common, but since the currents are low, one can use a switching regulator to get to 5 V and then use a power supply splitter IC to create the ground in the center.
With the right reference and choice of resistors, you should easily be able to achieve 100-200 nA. Printed Circuit Board layout at these levels is critical.
The LM10
**broken link removed** would probably be a good choice for the reference to set the current.
So, with a few parts, you have a current source that meets your needs with a +-2.5 V compliance (give or take)
This is a possible "rail splitter"
https://www.electro-tech-online.com/custompdfs/2011/06/tle2426-ep.pdf. You'll need a DC-DC converter to get 5V. Possibly 9 VDC to 5 VDC.
This one
**broken link removed** used an input of 2V to 3.6 volts.
You now have a basis for some sort of design. What's missing is how to set the compliance voltage and indicate out of compliance. This is where it gets tricky and I need to think some more.
At the very least, you'll need fixed references for the two voltage limits, again using LM10's.
A question then arises as to what to do when the voltage limit is exceeded?
1. disconnect
2. disconnect after a certain amount of time in the condition.
3. Reduce the current, so the voltage is in the required range. Indicate that the circuit has dome so.
4. Just indicate that it's out of range. LED, Buzzer etc.
(3) would be my choice.
I told you the design is not simple.