Help with complicated battery issues.

Some time ago my Model A1181 Macbook started to refuse to charge its battery. I have replaced the battery (new) and the battery interface cable/board (used) to no effect.

The magsafe connector is almost always green. It is not green for about a minute after boot or a minute after I plug in the battery. It thinks for this minute and then decides the battery does not exist. Black X. Despite this, starting with a running laptop plugged in to AC power with a charged battery.... the computer draws power (runs happily) from the battery it doesn't think exists when you unplug the AC power. I have tried all of the various resets and battery updates.

Is there anything which could be broken on the motherboard or hacked to become fixed?

From system profiler during the minute where its thinking about whether the battery exists:

Battery Information:

Charge Information:

Charge remaining (mAh): 0

Fully charged: No

Charging: Yes

Full charge capacity (mAh): 0

Health Information:

Cycle count: 0

Condition: Normal

Battery Installed: Yes

Amperage (mA): 0

Voltage (mV): 0