Appears that both supplies have an externally asserted,
hard turnoff (maybe the battery-drop, but maybe some
other "feature" of the power supply controller(s).
A question right off the top is, when you add the "large
capacitor" does the discharge ramp look like
dV/dt=Ioperating/Cbigfat, or is it getting jammed to ground
by some external, much lower, impedance? If this is the
case then maybe you need not just a hold-up capacitor
but a smart disconnect from the shunting device and
supply, until your own discharge at your own chosen time.
Like a negative load switch that "opens" when battery has
gone out, before converter UVLO has been hit to initiate
the discharge itself (if that's the chain of events).