Amplifying current from a photodiode
So as you can see, the drone needs approximately 2W to fly, whereas my photodiode can supply 378mW max....


You are short of power. Nothing can fix that (at least, neither amplifiers, nor converters).

If you had a converter, even with 100% efficiency, you could trade voltage for current, but you could not change the power.

If you had an amplifier, that would need an external source of power.

The one way you can fix a shortage of power is storage. You still have to respect total energy, but if you charge a storage medium at low power for a long time, you can discharge it at high power for a short time. Unfortunately, any storage solution weighs something (capacitors, batteries, reversible fuel cell and gas tanks), and you're very restricted with how much extra weight a drone can carry. 

A small LiPo is probably your best bet. The drone could sit on the ground sunning itself for an hour, then leap into the air for a few minutes.