If the battery is near death will it provide the current required by
the upconverter? It's not magic and you can't get more power out of
the upconverter than you put in. I'm curious to know if you have
measured the performance with a "battery near death"?

An upconverter can't solve the available capacity problem. It can only
make it worse. (About 15 percent worse for the referenced converters).
They seem to be of no value except for powering equipment that will not
operate properly unless the available voltage is increased.

My prefered solution for emergencies is to have a handheld tranceiver
with an alkaline battery pack.