it first went bad after being ten years old, yes right out of warranty.
Sounds about right. Some go bad sooner, some will last longer.
we opted for a rebuilt battery for $2600 which lasted about four years.
The problem with "rebuilt" batteries is only the bad cells are replaced, so you wind up with several new cells and a bunch of old ones. You're lucky to have gotten four years out of it.
We are now on the second rebuilt battery which is now about six months old and still under the one year warranty. The car is experiencing the same symptoms that happened both times before the battery completely gave up
Not unexpected.
What can I do to show this battery is failing before the one year warranty is up?
A more comprehensive battery check needs to be performed by a hybrid expert.
However keeping a 19-year-old Ford hybrid going is probably beating a dead horse. You were fortunate to get 14 years of use out of the thing, time for something else. Those are not particularly known for long-term reliability.