Assuming the LED was enough, if you pulsed the LED with a 50% duty cycle you only have current flowing half the time. The peak power would be the same as DC but the overall average power would be lower.
Still, how would that be more efficient?
The duty cycle will only be 50% so it will be half as bright, meaning you gain nothing.
Let me explain my idea. I have need to control Character LCD (40x4) and it's backlight needs 300mA @ 4.2V. Because of this I decided that I need something that can handle more power than MCU can. I also wanted to make one board for another project which just requires small 16x2 display.
I do need to setup brightness of LCD to certain level that depends on surroundings but once it is set, it will stay like that for eternity. I decided to use 12V rail for backlight and to adjust it's voltage by mean of PWM which I created with 555 because it was available in my toolbox and because it can work on 12V.
Why not use a 5V rail for the backlight and save 58.2% of the power required from 12V?
Increasing the voltage just increases the losses, the power in the series resistor is just wasted as heat.
If you can get a 5V SMPs mains adaptor it would be ideal.
Since I do need to adjust brightness (although rarely) simple resistor is not really good for me - I wanted something that will use trimmer (not displayed in schematics). I was worried about stress that overdriving LED (cca 900mA when on (10 ohm resistor) and then some time off). That is why I am attempting this low-pass filter. My idea is to give 7V to this display (PWMed from 12V) and that voltage would suit fixed 10 ohm resistor just fine.
Then don't overdrive the LED, use a resistor that will give 300mA at the maximum possible supply voltage and check the datasheet to ensure that 300mA is the recommend operating current, not the absolute maximum rating.
PWM is good though because it means you won't need a high powered variable resistor in series with the LED which will cost more than a cheap power resistor and a MOSFET. Why not drop the 555 and do PWM with the MCU? You could then use buttons or a pot with the ADC?
If you really must use 12V and efficiency is important. . .
Do you have a free comparator available on the MCU?
You could use it to build a very simple SMPs with just an inductor, a MOSFET, a few resistors, a capacitor, a diode and an inductor - there's a MicroChip app. note about this somewhere I could find if you're interested.