I notice that the maximum recommended Virtex2pro Vbatt is 2.63V.
> Lithium cells put out more than that.
Many chips don't go to sleep mode until the VDD has dropped quite a bit.
See:
The 2V63 value might be the ideal value on their VDD/IDD graph, with most headroom for battery voltage droop, but lower voltages (e.g. 2V4) might work just as well.
Is the solution as simple as a series Si diode on the Vbatt line?
Textbooks show zero current up to Vf but I suspect Vf drop depends on current, even before then.
It should be a quick experiment to see what happens.