How to supply more than 500mA while powered by USB
Do not bypass the fuse. Drive your external components directly from the external 5V power supply. Do not pass everything through the Arduino.

You say you have a regulated 5V power supply. Assuming it provides enough current you should use it to drive your external components directly.

You might need to add a filter capacitor near the feed for the Pi/Arduino, to filter out fluctuations in the 5V supply as your high current components switch on and off. You should be able to google "Power supply filter capacitor" to learn more.