I can rebind reload to 'P', but in game pressing it doesn't do anything. Binding reload to any of the other keys I've tried works (G, R, D). No other actions are bound to P.

Go to:

"%LOCALAPPDATA%\Insurgency\Saved\SaveGames\Profile"

And remove "Controls.json", then redo your in-game keybinds. Or edit the file and remove the bind for "P". That should fix it.