Check and write down the voltages measured at the following places when you are cranking the starter, with the negative meter lead attached to the negative battery terminal (not the screw or cable connector) and the positive lead attached to:
1 - The positive battery terminal (not the screw or connector)
2 - The positive battery connector
3 - The negative battery connector
4 - both sides of the starter solenoid / relay
5 - 6 - Both sides of the fuse (only one is active, the other is spare)
Then repeat the same measurements using somewhere on the FRAME as the grounding point for the negative meter lead. (7-12)
This may be just a start. I believe that somewhere in your electrical system you may have a high resistance connection, causing a voltage drop when cranking the starter, with enough of a drop to prevent the CDI from operating properly.
BTW, do these measurement with the battery alone, no charger attached , and only crank for a several seconds for each measurement.
Also, a quick check... when you crank the starter over in neutral, does the neutral light stay bright or dim significantly? I'm gonna guess it dims a lot.
'01 Scrambler 400, K&N
'02 400ex, piped, jetted, K&N, Lonestar +2 A-arms, Lonestar anti-vibe
'02 400ex, DiamondJ +3 A-arm widener, Lonestar anti-vibe
'02 660 Raptor, DiamondJ +2 A-Arm widener, YFZ shocks, LSR anti-vibe, piped, jetted, K&N