I built up the power loss protection circuit, made a few tweaks, and tried it with a very modest load (only 5ohms, i.e. 5W) and the poor AA batteries droop badly and cannot hold up anything like 4-4.5V. I'm getting more like 3.6V.

So maybe a change of tack? I have a little dc-dc converter based on the MP2307, which is supposedly good for 3A. It's hardwired to produce 5.0V at the moment, and therefore requires about 6.1V in. I'm thinking what I might do now is use a battery with more cells (possibly 6?) and the dc-dc converter. I can do away with the P-channel MOSFET since I can now use the enable line on the converter to shut off the power (leaving just 0.3uA standby current). I should now get much better supply regulation for the 10-20secs while the Pi shuts down and as the batteries droop.

I've bodged something together quickly on the bench, and it nearly works except that the TL431 cathode voltage doesn't rise fast enough to enable the dc-dc converter, before the rail supplying the TL431 falls and everything goes off. I could supply the TL431 direct from the battery, but that would flatten it in weeks.