I need to blink an LED (2.1V @350mA) @4Hz for ~30 seconds when an acceleration force is .5 gforce (+/ .1g) along one axis. The input (acceleration force) will last for 12 seconds. I have prototyped on an Arduino UNO and ADXL335 but now I am looking for a more cost effective solution.
I am thinking about using a tilt switch and 555 timer. I spec'd the tilt switch by converting gforce to degrees relative to the orientation of gravity (.5g = 30 degrees) . But I need help designing the 555 timer circuit.
I understand the basics of the 555 but things I am having trouble understanding are:

How to keep the LEDs blinking after zero input? Input is 12s for which the tilt switch circuit will be closed and I need to blink for 30s. Do I store energy in a larger capacitor from which the 555 circuit draws energy from?

How to spec the capacitor to power the LED? Once I know C, I can calculate R1 and R2 to achieve the blink frequency of 4Hz. Here's my thought process so far  the LED consumes 2.1V and 350mA = .735 Watts = .735 J/s And since the LED blinks at a rate of 4Hz, the energy needed for one blink is (.125s)(.735J /s) = .09 Joules. Then C= 2W/V2 and C=41mF. Does this make sense?? It seems off to me since that's a big cap.

What type of cap do I need? Aluminum Electrolytic , ceramic, etc..
I am an ME so I'd appreciate any ideas, thoughts, or direction you can provide in order to achieve the most cost effective solution. Thanks!