Hi,

My first idea was that the overcurrent protection function in the battery 
was activated (in both), surprising that happens with both but consistent 
with the fact that the machine works with the AA pack.

But this is not consistent with the fact that your machine does not work 
without the battery. I tried mine and it works. So the overprotection may be 
an issue but it is something else wrong with your machine.
When using the AA pack why not try the diagnostic utility?(Run 
\windows\ibmdiag.exe).
Question: you said the led stays always green, that means even without the 
ac power connected? I do not think so.
Question 2: have you contacted the seller? Seems a case of DOA that ebay and 
paypal should deal with.
Finally try leaving the machine alone for 24 hs without battery and back-up 
batteries.
Good luck