Electronic – Measure power usage of battery powered device

No, like you already may have feared this isn't kosher. The 3 V is indeed the minimum, and the AA batteries' voltage will quickly drop below that. If you're using NiMH rechargeables you even get only 2.4 V, so that won't do, unless you can use 3 of them in series. (Don't use batteries in parallel like you mention in your question.) So three NiMH cells will give you 3.6 V. Fine.

That 3.6 V is the maximum VDD for the device, so if you want to run it off USB power you'll need an LDO (Low Drop-Out) voltage regulator to get 3.6 V. The LP2981 is a good part for this.

Now the switching. 175 µs seems like an eternity to me, but we'll have to live with that. Ben gave you already the right equation for a constant current discharge:

\$ \Delta V = \dfrac{I \cdot \Delta t}{C} \$

or

\$ C = \dfrac{I \cdot \Delta t}{\Delta V} \$

NiMH cells have a fairly constant 1.2 V, which only drop to below 1.1 V when they're nearly discharged.

So we can use that as a limit. With a minimum voltage of 3 V and a worst-case current of 70 mA you get

\$ C = \dfrac{70 mA \cdot 175 \mu s}{300 mV} = 41 \mu F \$

which is what Ben also found. If you think you won't go below 1.15 V then that would become 27 µF, so that's not going to change very much, but it gives you some headroom if you want to use a 47 µF cap. AndrejaKo rightly points out that electrolytic capacitors have large tolerances, usually -20 %, and then I would just go for a 68 µF/6.3 V cap.

I think you can fix it as follows:

Put a pulldown on sysoff, maybe 100k. Just make sure it is strong enough to overpower the internal 5M pullup. Also keep sysoff connected to your IO expander. It should be configured as an output driven low. But when you want to power down, have the expander drive the output high, which will overpower the 100k pulldown, and turn the system off.

Once the system is off, your rails will all collapse, so the IO expander will be dead, and the 100k pulldown will ensure that sysoff is now low so that startup will be possible the next time charger is attached. You will need to make sure that the IO never goes high during bootup or normal operation. For example, program it to be low before you program it to be an output.