extra battery for 2 amplifiers?

Let us know what amps you have and what the included fusing is (i.e. 40A fuse on the 4-channel, etc.) You're probably safe using the factory battery only. There are several of us out here running up to 1,000 Watts of the factory setup with no issues.
I'd limit the running the stereo in Aux mode though. Why wouldn't you just have the car on? that way the engine can kick on and supply the needed voltage/current as needed.
I wouldn't want to have to figure out how integrate a second battery with a dual battery isolator in the Prius...the voltage regulation between the high voltage battery, electric motor/generator, 12V battery, and voltage inverter is fairly complex....there's a fusable link that could pop if too much current were drawn through.

 

I would chose to upgrade the stock battery with an Optima or somehting similar. The stock battery is pretty bad but you don't need two.

As for play time with the car off. It depends on the volume level. My system won't play at loud volumes with the car "off" due to a lack of voltage so the amp cuts in and out if I try it. Just leave the car in ready mode and play it. The ICE will come on as needed. I still don't recommend playing at max volume for long periods with the car off unless you plan on adding a large Optima (deep cycle) battery inline.

 

Recommend class D amp for the subs, they are the most efficient as far as using only as much power as needed to play your music.

I'm running an Alpine PDX 2.150 bridged for my sub (class D), Alpine PDX 4.100 for the bi-amped rear mids and tweets (class D), and a JL Audio 4/150 for the front mids and tweets (class a/b), all off the stock battery with no issues so far.

I don't sit listening unless I'm stuck in traffic.

Probably the only accurate way to know how long you could play sitting without the ICE running is to get an idea of how many amp-hours the system uses at a normal listening level. Then compare that with the amp-hour battery rating.

Practically, you could start with a fully charged battery, play your tunes 'till the amp(s) quit. Have someone give you a jump start (if you need one).

There are devices out there too called battery minders that will disconnect the battery from the circuit just before the voltage level drops to a non-start level. Probably cheaper to carry one of those booster packs, they work good as flashlights too.

While I can see I nice spot for a second battery, I agree with geogeek, it's a highly integrated power system that may get cranky if fiddled with.

Good luck.