What is the speed of the external alternator?

As just a Back-of-the-Envelope approximation...

A typical Auto A/C Compressor takes about 4 hp to run.

4 hp is ~3000 watts.  

So assuming the three Prius Sub-Units are 1000 watts Each; at 24 volts that would be ~40 amps Each!

To put that in perspective; a 3000w A/C compressor would be the equivalent load of a 240amp, 12volt Alternator.  Outch!

Probably WAAAY!!! more than you would want to couple to the Accessory drive gear on the 912 Gearbox.

It you are dead set on powereing an A/C Unit in flight; the only realistic option would be to mechanically couple to the engine via a belt drive type system.

Remember that if the Compressor were ever to seize up, you will NOT want it hard coupled to the engine. 

Depending on the size (+efficiency) of the compressor 1 kW of power would be enough for perhaps 5000 to 10000 BTU which is more than I would ever need so I'm not worried about the compressor putting too much of a load on the gearbox. How much is the accessory drive gear manufactured to handle? Could it handle 1 hp?

There used to be FlyCool https://www.bydanjohnson.com/video/flycool-air-conditioning-by-amt/ which is an electric unit that can be powered by the external alternator but they have left the light aircraft market according to Dan Johnson. That is how I know that the accessory drive gear is able to handle enough power to make the flight more cool and comfortable even if it is not as much as the Toyota Prius units.

PJ Asked:

How much is the accessory drive gear manufactured to handle? Could it handle 1 hp?

The Gear was intended to drive either a Vacuum Pump or a Hydraulic Propeller Govenor.

It has been used by third parties to drive 200-300 watt generators.

You are asking if it can handle three times that amount.

Maybe YES, Maybe NO!

You really are in uncharted trerritory here.

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Note in the Video above, The rep stated that it runs off a dedicated 24 volt, 1.5hp Alternator.  (1100 Watt, 45amp).