Ignoring the regulator, have the pushbutton supply input to the regulator.
Once the pic wakes up, it can turn on a PNP transistor, which is in
paralell with the button, to keep it's power up as long as it likes. When
it's done, just turn off the transistor.  This will take a pair of
transistors, but that's only pennies, and you won't have any availability
problems.

Now, solve the regulator as a separate problem.
Since it's input is cut off most of the time, its efficiency becomes much
less important.
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