To measure is to know.
Guessing does not help.

Get a workshop manual.
(At least this part)
And use your multi meter.

And be systematic.
Measure coil by coil. CDI box if possible.

The CDI system does not use battery.
It uses a two specific coils on the generator.
My money is on that the problem is there.
A simple resistance measurement will give the answer.

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The very old Kawasaki H2 had a CDI system.
With two coils. High speed and low speed.

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There is one test than can be done, besides measurements.
Try to push start the bike. At as a high speed as possible.
Than the high speed coil starts to work.
And the engine will be running as long as you keep it at high rpm.
But stop when you let it down to idle.

That would be the proof that the low speed coil is bad.

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Let us know the result. So we can improve our knowledge.
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You wrote that you had measured on the three pin connector. And all gave same value.
Strange.
Measure on the side towards the engine.
Between to pins at the time
The three reading will show values for
a) One coil
b) The other coil
c) The two coils in series

Look up the values in a works shop manual.
On another bike that I was working on the values were
5 Ohm
200 Ohm
For the two coils

Measure each of the three pins toward earth.
I shall be open circuit. Else there is a short circuit.

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The picture

Green: Coils to feed the CDI system with current
Yellow: Coil to trigger the ignation
Red: Charging the battery

Blue 1 and 2: The high and low speed coils.

Attached Images