2. Now, instead of the LED, I wired the handset, connected the
battery ( 3.6V, 300mAH) to the battery terminals on the handset
PCB.
The charge/in-use LED of the base unit lit up... but after
about a minute, the LED of the base unit is no more on...
Removed the battery, measured the terminal voltage and it
shows 3.6V but a little bit usage ( by pressing talk button on the
handset) drops it to 2.2V
The battery pack is brand new and not a old one so I cannot doubt the
battery.

You're assuming that the light going out means the battery has completely
charging. This might not be true. Let it sit longer.


So basically, I still have not fixed this instrument. How can
the battery voltage jump to 3.6V in just a minute ?

It's normal for nickel-cadmium batteries to return to "full" (or near-full)
voltage after a short charge -- but the battery _isn't_ fully charged. Be
patient.