Are there any high energy density (i.e. zinc-air or lithium-manganese dioxide) batteries without high internal resistance?
Tadiran Batteries have LiSOCl with an integrated super capacitors. Those are not rechargeable and are quite expansive compared to the more common CR types.
The standard LiSOCl AA cell can manage 100mA/100mSec pulse at ~3V (Dropping from 3.6V).
There are many manufacturers of such batteries (Tadiran, Saft, Xeno, EEMB and others).
Of course, there are also D sized cells with 19Ah capacity and much higher pulse capabilities.