Well, divide 11.4V by the LEDs Max Vforward, the whole number part will tell you how many can go in a string.
Then figure the current limit resistor = (11.4V - #LEDs * Minimum Vforward)/20mA, use Minimum Vforward so you can never exceed the 20mA.
Then go search Digikey for N-Channel Mosfet,
select Single FETs
then start filtering:
> In stock
> through hole
> logic level
that should narrow the list a lot
Click on View page
Click on sort by price, then browse down to find the Lowest Rds you can find (down in the low mOhm kind of range) in a package you can work with at a price you like. At 20mA per string of LEDs, you can have a lot of strings switched by 1 really low Rds MOSFET before it starts to gets warm.
P = I*I*R.
So with your supply and 12mOhm Rds for example
P = 3.95A * 3.95 * .012 = 187mW which may not even need a heatsink since you are strobing & not on all the time.
That's assuming you have 3.95A/20mA/string = 197 strings of LEDs! In which case you better hand out shade too