What is the best program to make short videos, say around 30 seconds max, in P3D that wouldn't drag down the FPS?   

which I used several times was the ACTION

Nvidia ShadowPlay if you have an supported Nvidia GPU

nvidia shadowplay as stated. Assign a button on your flight controls to the keyboard shortcut of shadowplay and you can record on the fly. No FPS hit. Shadowplay is hidden in the nVidia Experience software. You need to activate it manually and you have to tag "Desktop Capture" You can further define the quality and the file splitting if you like.