AceofWWII:
Constraints: The solution needs to be as compact as humanly possible, so the fewer items, the better!
Then don't use an Arduino, not even the still bulky Pro Mini, but an ATtiny25/45/85 or other 8-pin AVR, preferably in VSSOP package. All you have to add is a regulator (SOT23-5 SMD package is tiny), filter cap (can come in tiny package - like the 0604 SMD) and indeed a MOSFET for switching (get one in SOT-23 package) and a resistor to protect your pin (can also be 0604 size).
Making it like this will make your LED look big.